978-3-98547-561-2-volume-3

INTERNATIONAL GONGRESS

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To describe the constant coefficients of

E_{k} (χ, 1)

we first set

S

to be minimal, i.e., the union of

S_{\infty}

and

S_{ram}

, where the latter is the set of primes dividing

n

. Next we assume for the remainder of the article that

n \neq 1

; the case

n = 1

causes no difficulties but the formulas must be slightly modified. We then have

c_{λ} (0, E_{1} (χ, 1)) = {\begin{cases} 0, & k > 1 \\ 2^{- n} Θ_{S}^{#}, & k = 1 \end{cases}

Here

Θ_{S} = Θ_{S, ϕ} (H / F, 0) \in Q [G]

denotes the

S

-depleted but unsmoothed Stickelberger element. We have

E_{k} (χ, 1) \in M_{k} (n, χ; R_{p})

for

k > 1

and

E_{1} (χ, 1) \in M_{1} (n, χ; Frac (R_{p}))

because of the possible nonintegrality of the constant term.

6.4. Eisenstein series to cusp forms

In order to define a cusp form from the Eisenstein series, one is led to consider certain linear combinations of the analogues of

E_{k} (χ, 1)

H

ranges over all its CM subfields containing

F

. This process also incorporates smoothing at the primes in

T

. We avoid stating the slightly complicated formula here (see [17, PRoPoSITION 8.14]), but the end result is a group ring form

W_{k} (χ, 1)

whose constant terms are given by

\begin{matrix} (6.12) & c_{λ} (0, W_{k} (χ, 1)) = {\begin{cases} 0, & k > 1 \\ 2^{- n} Θ^{#}, & k = 1 \end{cases} \end{matrix}

where we remind the reader that

Θ^{#} = Θ_{Σ, Σ^{'}}^{#}

. Building off the computations of [19], we calculate in

[17, $ 8]

the constant terms of the

W_{k} (χ, 1)

at all cusps; the terms in (6.12) can be viewed as the constant terms "at infinity." Indeed, it is the attempt to cancel the constant terms at other cusps that leads naturally to the definition of the

W_{k} (χ, 1)

In order to define a cusp form, we apply two important results of Silliman [43]. The first of these generalizes a result of Hida and Wiles and is stated below.

Theorem 6.2 ([43, THEOREM 10.7]). Let

m

be a fixed positive integer. For positive integers

k \equiv 0 (mod (p - 1) p^{N})

with

N

sufficiently large, there is a Hilbert modular form

V_{k}

of level 1, trivial nebentypus, and weight

k

defined over

Z_{p}

such that

V_{k} \equiv 1 (mod p^{m})

and such that the normalized constant term of

V_{k}

at every cusp is congruent to

1 (mod p^{m})

The idea to construct a cusp form is to fix a very large integer

m

and to consider the product

W_{1} (χ, 1) V_{k} \in M_{k + 1} (n, χ, R_{p})

with

V_{k}

as in Theorem 6.2. This series has constant terms at infinity congruent to

2^{- n} Θ^{#}

modulo

p^{m}

. One then wants to subtract off

2^{- n} Θ^{#} H_{k + 1} (χ)

for some group ring valued form

H_{k + 1} (χ) \in M_{k + 1} (n, χ, R_{p})

to obtain a cusp form. If there exists a prime above

p

dividing

n

(i.e.,

Σ - S_{\infty}

is nonempty), then this strategy works. Silliman's second result, which generalizes a result of Chai and is stated in [43, THEOREM 10.10], implies that one can obtain a form that is cuspidal at the cusps "above infinity at

p

" in this fashion. Applying Hida's ordinary operator then yields a form that is cuspidal.

Theorem 6.3 ([17, theOREM 8.18]). Suppose

\gcd (n, p) \neq 1

. For positive integers

k \equiv 1

(mod (p - 1) p^{N})

and

N

sufficiently large, there exists

H_{k} (χ) \in M_{k} (n, χ, R_{p})

such that

F_{k} (χ) = e_{p}^{ord} (W_{1} (χ, 1) V_{k - 1} - Θ^{#} H_{k} (χ))

lies in

S_{k} (n p, R, χ)

The significance of Theorem 6.3 is that we have now constructed a cusp form that is congruent to an Eisenstein series modulo

Θ^{#}

When

\gcd (n, p) = 1

, the construction of the cusp form is in fact more interesting, and a new feature appears. In this case, the ordinary operator at

p

does not annihilate the form

W_{k} (χ, 1)

, and it must be incorporated into our linear combination. Moreover, this apparent
cost has a great benefit-we obtain a congruence between a cusp form and Eisenstein series not only modulo

Θ^{#}

, but modulo a multiple

x \cdot Θ^{#}

for a certain

x \in R_{p}

. This element

x

has an intuitive meaning-it represents the trivial zeroes of the

p

-adic

L

-function associated to

χ

, even the "

mod p

trivial zeroes." The precise definition is as follows.

Lemma 6.4. Suppose

\gcd (n, p) = 1

. For positive

k \equiv 1 (mod (p - 1) p^{N})

with

N

sufficiently large, the element

x = \frac{Θ_{S_{\infty}} (1 - k)}{Θ_{S_{\infty}} (0)} \in Frac (R)

lies in

R

and is a non-zero-divisor.

The analogue of Theorem 6.3 for

\gcd (n, p) = 1

is as follows.

Theorem 6.5 ([17, theorem 8.17]). Suppose

\gcd (n, p) = 1

. For positive integers

k \equiv 1

(mod (p - 1) p^{N})

and

N

sufficiently large, there exists

H_{k} (χ) \in M_{k} (n, χ, R_{p})

such that

F_{k} (χ) = e_{p}^{ord} (x W_{1} (χ, 1) V_{k - 1} - W_{k} (χ, 1) - x Θ^{#} H_{k} (χ))

lies in

S_{k} (π p, R, χ)

The extra factor of

x

in our congruence between the cusp form

F_{k} (χ)

and a linear combination of Eisenstein series plays an extremely important role in showing that the Galois cohomology classes we construct are unramified at

p

We conclude this section by interpreting the congruences of Theorems 6.3 and 6.5 in terms of Hecke algebras. We consider the Hecke algebra

\tilde{T}

generated over

R_{p}

by the operators

T_{q}

for primes

q ∤ n p

and

U_{p}

for primes

p ∣ p

. (We ignore the operators

U_{q}

for

q ∣ n, q ∤ p

in order to avoid issues regarding nonreducedness of Hecke algebras arising from the presence of oldforms.) We denote by

T = e_{p}^{ord} (\tilde{T})

Hida's ordinary Hecke algebra associated to

\tilde{T}

. Let

ϵ_{cyc} : G_{F} \to Z_{p}^{*}

denote the

p

-adic cyclotomic character of

F

. Theorems 6.3 and 6.5 then yield:

Theorem 6.6. Let

x = 1

\gcd (n, p) \neq 1

and let

x

be as in Lemma 6.4 if

\gcd (n, p) = 1

. There exists an

R_{p} / x Θ^{#}

-algebra

W

and a surjective

R_{p}

-algebra homomorphism

φ : T \to W

satisfying the following properties.

The structure map $R_{p} / x Θ^{#} \to W$ is an injection.
$φ (T_{Y}) = ϵ_{cyc}^{k - 1} (l) + χ (l)$ for $n p$ .
$φ (U_{p}) = 1$ for $p ∣ \gcd (n, p)$ .
Let

U = \prod_{p ∣ p, p ł n} (U_{p} - χ (p)) \in T

y \in R_{p}

and

φ (U) y = 0

W

, then

y \in (Θ^{#})

The idea of this theorem is the usual one: the homomorphism

φ

sends a Hecke

point: the operators

U_{p}

for

p ∣ p, p ∤ n

do not act as scalars, so a more involved argument is necessary. This explains why the ring

W

is not just

R_{p} / x Θ^{#}

. The idea behind the last statement of the theorem is that the operator

φ (U)

introduces a factor of

x

; If

x y

is divisible by

x Θ^{#}

R_{p}

, then

y

is divisible by

Θ^{#}

since

x

is a non-zero-divisor. This demonstrates the essential additional ingredient provided by the "higher congruence" modulo

x Θ^{#}

rather than just modulo

Θ^{#}

. See [17, THEOREM 8.23] for details.

6.5. Cusp forms to Galois representations

In this section we study the Galois representation attached to cusp forms that are congruent to Eisenstein series. Let

m

be the intersection of the finitely many maximal ideals of

T

containing the kernel of

φ

. Put

T_{m}

for the completion of

T

with respect to

m

and

K = Frac (T_{m})

. Then

K

is a finite product of fields parameterized by the

Q_{p}

-Galois orbits of cuspidal newforms of weight

k

and level

n

, defined over the ring of integers in a finite extension of

Z_{p}

, that are congruent to an Eisenstein series modulo the maximal ideal. As in [17, $9.2], the work of Hida and Wiles gives a Galois representation

ρ : G_{F} \to {G L}_{2} (K)

satisfying the following:

(1)

ρ

is unramified outside

n p

(2) For all primes

l ∤ n p

, the characteristic polynomial of

ρ

(Frob

l)

is given by

\begin{matrix} (6.13) & char (ρ ({F r o b}_{l})) (x) = x^{2} - T_{r} x + χ (l) {N I}^{k - 1} \end{matrix}

(3) For every

q ∣ p

, let

G_{q}

denote a decomposition group at

q

. We have

{ρ |}_{G_{q}} \sim (\begin{array}{cc} χ ε_{c y c}^{k - 1} η_{q}^{- 1} & * \\ 0 & η_{q} \end{array})

where

ε_{cyc}

is the

p

-adic cyclotomic character and

η_{Q}

is an unramified character given by

η_{q} ({rec}_{q} (ϖ^{- 1})) = U_{q}

, with

ϖ

a uniformizer of

F_{q}^{*}

Let I denote the kernel of

φ

extended to

φ : T_{m} \to W

. Reducing (6.13) modulo

I

, and using ÄŒebotarev to extend from Frob

b_{I}

to all

σ \in G_{F}

, we see that the characteristic polynomial of

ρ (σ)

is congruent to

(x - χ (σ)) (x - ϵ_{c y c} (σ)) (mod I)

. In particular, if

χ (σ) ≢ ϵ_{c y c} (σ) (mod m)

, then by Hensel's lemma

ρ (σ)

has distinct eigenvalues

λ_{1}, λ_{2} \in T_{m}

such that

λ_{1} \equiv ϵ_{cyc}^{k - 1} (σ) (mod I)

and

λ_{2} \equiv χ (σ) (mod I)

To define a convenient basis for

ρ

, we choose

τ \in G_{F}

such that:

(1)

τ

restricts to the complex conjugation of

G

(2) for each

q ∣ p

, the eigenspace of

{ρ |}_{G_{q}}

projected to each factor of

K

is not stable under

ρ (τ)

See [17, PRopositIoN 9.3] for the existence of such

τ

. Since

p \neq 2

, we have

χ (τ) = - 1 ≢ 1 \equiv ϵ_{c y c} (τ) (mod m)

It follows from the discussion above that the eigenvalues of

ρ (τ)

satisfy

λ_{1} \equiv ϵ_{c y c} (σ) (mod I)

and

λ_{2} \equiv - 1 (mod I)

. Fix the basis consisting of eigenvectors of

ρ (τ)

, say

ρ (τ) = (\begin{array}{cc} λ_{1} & 0 \\ 0 & λ_{2} \end{array})

For a general

σ \in G_{F}

, write

ρ (σ) = (\begin{array}{ll} a (σ) & b (σ) \\ c (σ) & d (σ) \end{array})

For each

q ∣ p

, there is a change of basis matrix

M_{q} = (\begin{array}{ll} A_{q} & B_{q} \\ C_{q} & D_{q} \end{array}) \in {G L}_{2} (K)

satisfying

\begin{matrix} (6.14) & (\begin{array}{ll} a (σ) & b (σ) \\ c (σ) & d (σ) \end{array}) M_{q} = M_{q} (\begin{array}{cc} χ ε_{c y c}^{k - 1} η_{q}^{- 1} & * \\ 0 & η_{q} \end{array}) \end{matrix}

The second condition in the choice of

τ

ensures that

A_{q}, C_{q} \in K^{*}

. Furthermore, equating the upper left-hand entries in (6.14) gives

\begin{matrix} (6.15) & b (σ) = \frac{A_{q}}{C_{q}} (a (σ) - χ ε_{c y c}^{k - 1} η_{q}^{- 1} (σ)) for all σ \in G_{q} \end{matrix}

6.6. Galois representations to Galois cohomology classes

We summarize [17, Â§9.3]. As explained above, we have

\begin{matrix} (6.16) & a (σ) + d (σ) \equiv χ (σ) + ϵ_{c y c}^{k - 1} (σ) (mod I) for all σ \in G_{F} \end{matrix}

Applying the same rule for

τ σ

and noting that

a (τ σ) = λ_{1} a (σ), d (σ τ) = λ_{2} d (σ)

, we find

\begin{matrix} (6.17) & a (σ) ϵ_{c y c}^{k - 1} (τ) - d (σ) \equiv - χ (σ) + ϵ_{c y c}^{k - 1} (σ τ) (mod I) \end{matrix}

Solving the congruences (6.16) and (6.17) and once again using the fact that

ϵ_{c y c}^{k - 1} (τ) ≢ - 1

(mod m)

since

p \neq 2

, we find that

a (σ), d (σ) \in T_{m}

and

\begin{matrix} (6.18) & a (σ) \equiv ε_{c y c}^{k - 1} (σ) (mod I), d (σ) \equiv χ (σ) (mod I) for all σ \in G_{F} \end{matrix}

Let

B

be the

T_{m}

submodule of

K

generated by

{b (σ) : σ \in G_{F}} \cup {\frac{A_{q}}{C_{q}} : q \in

Σ ∖ S_{\infty}}

. We have

ρ (σ σ^{'}) = ρ (σ) ρ (σ^{'})

for

σ, σ^{'} \in G_{F}

. Equating the upper right entries and using equation (6.18), we obtain

\begin{matrix} (6.19) & b (σ σ^{'}) = a (σ) b (σ^{'}) + b (σ) d (σ^{'}) \equiv ε_{c y c}^{k - 1} (σ) b (σ^{'}) + χ (σ^{'}) b (σ) (mod I B) \end{matrix}

Let

m

be am integer such that

k \equiv 1 (mod (p - 1) p^{m})

. Let

I_{q}

denote the inertia subgroup of

G_{F}

of a prime q. Put

B_{1}

for the

T_{m}

-submodule of

B

generated by

I B \cup p^{m} B \cup {b (σ) : σ \in I_{q} for q ∣ p, q \notin Σ}

Define

\bar{B} = B / B_{1}

. Equation (6.19) then gives that

κ (σ) = χ^{- 1} (σ) b (σ)

is a cocyle defining a cohomology class

[κ]

H^{1} (G_{F}, \bar{B} (χ^{- 1}))

satisfying the following local properties:

(1) As

ρ

is unramified at

Υ ∤ π p

, so is the class

[κ]

(2) As

\bar{B}

is pro-

p

, the class

[κ]

is at most tamely ramified at any prime

l ∣ n

not above

p

(3) It is proven in

[17, 84.1]

that we may assume

Σ^{'}

does not contain any primes above

p

. Thus

[κ]

is at most tamely ramified at all primes in

Σ^{'}

(4) By the definition of

B_{1}

, where we have included

b (I_{q})

for primes

q ∣ p, q \notin Σ

, the class

[κ]

is unramified at such

q

(5) Equation (6.15) implies that

[κ]

is locally trivial at finite primes in

Σ

. As

p

is odd,

[κ]

is locally trivial at archimedian places [17, PROPOSITION 9.5].

6.7. Galois cohomology classes to class groups

The Galois cohomology class

[κ]

satisfies the conditions listed after equation (6.2) and hence gives a surjection

\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-} \to \bar{B} (χ^{- 1})

For details see [18, THEOREM 4.4]. The general properties of Fitting ideals imply

{Fitt}_{R_{p}} (\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-}) \subset {Fitt}_{R_{p}} (\bar{B} (χ^{- 1}))

It is therefore enough to prove that

{Fitt}_{R_{p}} (\bar{B}) \subset (Θ^{#})

. Typically in Ribet's method, one argues that the fractional ideal

B

is a faithful

T_{m}

-module, and hence the Fitting ideal of

B / I B

is contained in

I

. However, our module

\bar{B}

is more complicated than

B / I B

, so we proceed as follows. Using equation (6.15), we show that any element in

{Fitt}_{R_{p}} (\bar{B})

is annihilated by

φ (U)

for the operator

U

from Theorem 6.6. The final assertion of this theorem then implies that

{Fitt}_{R_{p}} (\bar{B})

contained in

(Θ^{#})

. See [17, Â§9.5] for details.

This concludes our summary of the proof of Theorem 5.6.

7. EXPLICIT FORMULA FOR BRUMER-STARK UNITS

In this final section of the paper, we discuss the first author's explicit formula for Brumer-Stark units as mentioned in

§ 4.3

. The conjecture in the case that

F

is a real quadratic field was studied in [13], and the general case was studied in [15]. Here we consider an arbitrary totally real field

F

, but to simplify formulas we assume that the rational prime

p

is inert in

F

. Furthermore, we let

H

be the narrow ray class field of some conductor

n \subset O_{F}

and assume that

p \equiv 1 (mod n)

. This ensures that the prime

p = p O_{F}

splits completely in

H

. Fix a prime

P

H

above

p

. We fix

S \supset S_{\infty} \cup S_{ram} = {v ∣ n \infty}

. We also fix a prime ideal

l \subset O_{F}

such that

N Y = ℓ > n + 1

is a prime integer and let

T = {l}

In this setting, we will present a

p

-adic analytic formula for the image of the BrumerStark unit

u_{p} \in H^{*}

H_{P}^{*} ≅ F_{p}^{*}

. The most general, conceptually satisfying, and theoretically useful form of this conjecture uses the Eisenstein cocycle. This is a class in the

(n - 1)

st cohomology of

{G L}_{n} (Z)

that has many avatars studied by several authors (see

[2, 3, 9, 10, 13

20, 45]). In this paper, we avoid defining the Eisenstein cocycle and present instead the more explicit and down to earth version of the conjectural formula for

u_{p}

stated in [15].

7.1. Shintani's method

Fixing an ordering of the

n

real embeddings of

F

yields an map

F ↪ R^{n}

such that the image of any fractional ideal is a cocompact lattice. We let

F^{*}

act on

R^{n}

by composing this embedding with componentwise multiplication and denote the action by

*

Let

v_{1}, \dots, v_{r} \in {(R^{> 0})}^{n}, 1 \leq r \leq n

, be vectors in the totally positive orthant that are linearly independent over

R

. The corresponding simplicial cone is defined by

C (v_{1}, \dots, v_{r}) = {\sum_{i = 1}^{r} t_{i} v_{i} : 0 < t_{i}} \subset {(R^{> 0})}^{n}

Suppose now

r = n

. We will define a certain union of

C (v_{1}, \dots, v_{n})

and some of its boundary faces that we call the Colmez closure. Write

(0, 0, \dots, 1) = \sum_{i = 1}^{n} q_{i} v_{i}, q_{i} \in R

For each nonempty subset

J \subset {1, \dots, n}

, we say that

J

is positive if

q_{i} > 0

for all

i \notin J

. The Colmez closure of

C (v_{1}, \dots, v_{n})

is defined by:

C^{*} (v_{1}, \dots, v_{n}) = ⨆_{J positive} C ({v_{j}, j \in J})

Let

E (n) \subset O_{F}^{*}

denote the subgroup of totally positive units

ϵ

such that

ϵ \equiv 1

(mod n)

. Shintani proved that there exists a union of simplicial cones that is a fundamental domain for the action of

E (n)

{(R^{> 0})}^{n}

. For example, in the real quadratic case

(n = 2)

E (n) = ⟨ ϵ ⟩

is cyclic and

C^{*} (1, ϵ)

is a fundamental domain. In the general case, it can be difficult to write down an explicit fundamental domain, but a nice generalization of the

n = 2

case is obtained if we allow ourselves to consider instead a signed fundamental domain. For a simplicial cone

C

, let

1_{C}

denote the characteristic function of

C

{(R^{> 0})}^{n}

Definition 7.1. A signed fundamental domain for the action of

E (n)

{(R^{> 0})}^{n}

is by definition a formal linear combination

D = \sum_{i} a_{i} C_{i}

of simplicial cones

C_{i}

with

a_{i} \in Z

such that

\sum_{u \in E (n)} \sum_{i} a_{i} 1_{C_{i}} (u * x) = 1

for all

x \in {(R_{> 0})}^{n}

Fix an ordered basis

{ϵ_{1}, \dots, ϵ_{n - 1}}

for

E (n)

. Define the orientation

\begin{matrix} (7.1) & w_{ϵ} = sign \det {(\log (ϵ_{i j}))}_{i, j = 1}^{n - 1}) = \pm 1 \end{matrix}

where

ϵ_{i j}

denotes the

j

th coordinate of

ϵ_{i}

. For each permutation

σ \in S_{n - 1}

let

v_{i, σ} = ϵ_{σ (1)} \dots ϵ_{σ (i - 1)} \in E (n), i = 1, \dots, n

By convention,

v_{1, σ} = (1, 1, \dots, 1)

for all

σ

. Define

w_{σ} = (- 1)^{n - 1} w_{ϵ} sign (σ) sign (\det {(v_{i, σ})}_{i = 1}^{n}) \in {0, \pm 1}

The following result was proven independently by Diaz y Diaz-Friedman [22] and Charollois-Dasgupta-Greenberg [10, THEOREM 1.5], generalizing the result of Colmez [11] in the case that all

w_{σ} = 1

Theorem 7.2. The formal linear combination

\sum_{σ \in S_{n - 1}} w_{σ} C^{*} (v_{1, σ}, \dots, v_{n, σ})

is a signed fundamental domain for the action of

E (n)

{(R^{> 0})}^{n}

7.2. The formula

Throughout this section assume that

p

is odd. Recall that

T = {l}

. Let

b

be a fractional ideal that is relatively prime to

n l

. Let

D = \sum a_{i} C_{i}

be the signed fundamental domain for the action of

E (n)

{(R^{> 0})}^{n}

given in Theorem 7.2. We use all this data to define a

Z

-valued measure

μ

O_{p}

, the

p

-adic completion of

O_{F}

. Fix an element

z \in b^{- 1}

such that

z \equiv 1 (mod n)

. For each compact open set

U \subset O_{p}

, define the Shintani zeta function

ζ (b, U, D, s) = \sum_{i} a_{i} \sum_{\begin{matrix} α \in C_{i} \cap b - 1 \\ α \in U, (α, S) = 1 \end{matrix}} (N α)^{- s}

Shintani proved that this sum converges for

ℜ (s)

large enough and extends to a meromorphic function on

C

. Define

μ_{b} (U) = ζ (b, U, D, 0) - ℓ \cdot ζ (b L^{- 1}, U, D, 0)

Using Shintani's formulas, one may show:

Theorem 7.3 ([15, PRoPosition 3.12]). For every compact open

U \subset O_{p}

, we have

μ_{b} (U) \in Z

We may now state our conjectural exact formula for the Brumer-Stark unit

u_{p}

and all of its conjugates over

F

. Write

Θ_{S, T} = \sum_{σ \in G} ζ_{S, T} (σ) σ^{- 1}, ζ_{S, T} (σ) \in Z

Define

\begin{matrix} (7.2) & u_{p} (b)^{a n} = p^{ζ s, T (σ_{b})} ψ_{O_{p}^{*}} x d μ_{b} (x) \in F_{p}^{*} \end{matrix}

Here the crossed integral is a multiplicative integral in the sense of Darmon [12] and can be expressed as a limit of Riemann products:

f_{O_{p}^{*}} x d μ_{b} (x) := lim_{m \to \infty} \prod_{a \in {(O_{p} / p^{m})}^{*}} a^{μ_{F} (a + p^{m} O_{p})}

Write

σ_{b} \in G

for the Frobenius associated to

b

. In [15, THEOREM 5.15] we prove that

u_{p} (b)^{a n}

depends only on the image of

b

in the narrow ray class group of conductor

n

, i.e., on

σ_{b} \in G

(at least up to a root of unity in

F_{p}^{*}

Conjecture 7.4. We have

σ_{b} (u_{p}) = u_{p} (b)^{a n}

F_{p}^{*}

The expression (7.2) can be computed to high

p

-adic precision on a computer. See [23] for tables of narrow Hilbert class fields of real quadratic fields determined using this formula.

It is convenient to have an invariant that also satisfies

u_{p} (b q)^{a n} = {(u_{p} (b)^{a n})}^{- 1}

q

is a prime such that

σ_{q} = c

. Conjecture 7.4 would imply such a formula, but it is unclear whether this purely analytic statement can be proved unconditionally. To this end, we fix

q

such that

σ_{\dot{q}} = c

and define

v_{p} (b)^{a n} = {(\frac{u_{p} (b)^{a n}}{u_{p} (b q)^{a n}})}^{1 / 2} \in {\hat{F}}_{p}^{*} := F_{p}^{*} \hat{\otimes} Z_{p}

One then has

\begin{matrix} (7.3) & v_{p} (b q)^{a n} = {(v_{p} (b)^{a n})}^{- 1} \end{matrix}

unconditionally, and we expect to have

v_{p} (b)^{a n} = u_{p} (b)^{a n}

. The following is therefore a slightly easier form of Conjecture 7.4 to study.

Conjecture 7.5. We have

σ_{b} (u_{p}) = v_{p} (b)^{a n}

{\hat{F}}_{p}^{*}

7.3. Horizontal Iwasawa theory

We now discuss the relationship between Gross's tower of fields conjecture (Conjecture 4.4) and our conjectural exact formula for Brumer-Stark units. Our goal is to prove:

Theorem 7.6. Assume that

p

is odd. Gross's conjecture implies Conjecture 7.5.

In this exposition we have assumed that the odd prime

p

is inert in

F

and that

p = p O_{F}

. In the case of general

p

, one must still assume that

p

is odd and unramified in

F

in the statement of Theorem 7.6.

The abelian extensions

L / F

to which we can apply Gross's conjecture (with

S^{'} = S \cup {p}

) as in Conjecture 4.4) are those that contain

H

and are unramified outside

S^{'} \infty

. Let

F_{S^{'}}

denote the maximal abelian extension of

F

unramified outside

S^{'} \infty

. The reciprocity map of class field theory yields an explicit description of

Gal (F_{S^{'}} / H)

. For each finite

v \in S^{'}

, let

U_{v, n} \subset O_{v}^{*}

denote the subgroup of elements congruent to 1 modulo

n O_{v}

(so

U_{f, n} = O_{v}^{*}

for

v ∤ n

). Define

O^{*} = \prod_{v \in S^{'} ∖ S_{\infty}} U_{v, n}

. Then

Gal (F_{S^{'}} / H) ≅ O^{*} / \bar{E (n)}

where

\bar{E (n)}

denotes the topological closure of

E (n)

embedded diagonally in

O^{*}

For each finite extension

L \subset F_{S^{'}}

containing

H

, if we write

Γ = Gal (L / H)

, then (4.7) yields a formula for

{rec}_{G} (u_{p})

I / I^{2} ≅ Z [G] \otimes Γ

. Under this isomorphism, the coefficient of

σ_{b}^{- 1}

is just the image of

{rec}_{p} (σ_{b} (u_{p}))

Γ

. Taking the inverse limit over all

H \subset L \subset F_{S^{'}}

therefore gives an equality for

(σ_{b} (u_{p}), 1, 1, \dots, 1) in O / \bar{E (n)}

Here we have written

O_{p}^{*}

as the first component of

O

The next key point is that the constructions of Section 7.2 can be repeated to provide a measure

μ_{b, O}

O = \prod_{v \in S^{'} ∖ S_{\infty}} O_{v}

extending the measure

μ_{b}

O_{p}

. It is not hard to check that the restriction of

μ_{b, o}

O^{*}

, pushed forward to

O^{*} / \bar{E (n)}

, is precisely the measure that recovers the values of the partial zeta functions of the abelian extensions

L

contained in

F_{S^{'}}

. These are exactly the values appearing in Gross's conjecture. In other words, Gross's conjecture for the set

S^{'}

is equivalent to

\begin{matrix} (7.4) & (σ_{b} (u_{p}), 1, 1, \dots, 1) \cdot p^{- ζ_{S, T} (σ_{b})} = \int_{O^{*}} x d μ_{b, O} (x) in O / \bar{E (n)} \end{matrix}

See [15, PROPOSITION 3.4]. The next important calculation ([15, THEOREM 3.22]) is that

\begin{matrix} (7.5) & p^{ζ_{S, T} (σ_{b})} \int_{O^{*}} x d μ_{b, o} (x) = (u_{p} (b)^{a n}, 1, 1, \dots, 1) \end{matrix}

The first component of this is simply the compatibility of the constructions of

μ_{b}

and

μ_{b, o}

; the interesting part of the computation is the 1's in the components away from

p

. Equations (7.4) and (7.5) combine to yield that the ratio

σ_{b} (u_{p}) / u_{p} (b)^{an}

lies in the group

D (S) = {x \in O_{p}^{*} : (x, 1, 1, \dots, 1) \in \bar{E (n)} \subset O^{*}}

We can also conclude

\begin{matrix} (7.6) & σ_{b} (u_{p}) / v_{p} (b)^{a n} \in D (S) \end{matrix}

since

c (u_{p}) = u_{p}^{- 1}

The final trick, inspired by the method of Taylor-Wiles, is to consider certain enlarged sets

S_{Q} = S \cup Q

for a well-chosen finite set of auxiliary primes

Q

. Let us compare the Brumer-Stark units for

S

and

S_{Q}

, denoted

u_{p}

and

u_{p} (S_{Q})

, respectively. The defining property (4.1) shows that

u_{p} (S_{Q}) = u_{p}^{z}, where z = \prod_{q \in Q} (1 - σ_{q}^{- 1}) \in Z [G]

In particular, if we choose the

q \in Q

such that

σ_{q} = c

, the complex conjugation of

G

, then

u_{p} (S_{Q}) = u_{p}^{2^{2 \notin Q}}

. Using (7.3), one can similarly show that

v_{p} (S_{Q}, b) = v_{p} (b 2^{2^{2 Q}}

. Now (7.6) for

S_{Q}

implies that

σ_{b} (u_{p} (S_{Q})) / v_{p} {(S_{Q}, b)}^{a n} \in D (S_{Q})

hence

{(σ_{b} (u_{p}) / v_{p} (b)^{a n})}^{2^{# Q}} \in D (S_{Q}), so σ_{b} (u_{p}) / v_{p} (b)^{a n} \in D (S_{Q})

since

p \neq 2

To conclude the proof of Theorem 7.6, one shows using the ÄŒebotarev Density Theorem that one can choose the sets

Q

to force

D (S_{Q})

as small as desired (i.e., the intersection of

D (S_{Q})

over all possible

Q

is trivial). See [15, LEMMA 5.17] for details.

7.4. The Greenberg-Stevens $L$ -invariant

We briefly summarize our proof of the

p

-part of Gross's conjecture (Theorem 4.5), which as just explained implies our explicit formula for Brumer-Stark units given in Conjecture 7.5.

The work of Greenberg and Stevens [24] was a seminal breakthrough in the study of trivial zeroes of

p

-adic

L

-functions. Their perspective was highly influential in [16], where the rank one

p

-adic Gross-Stark conjecture was interpreted as the equality of an algebraic

L

-invariant

L_{alg}

and an analytic

L

-invariant

L_{an}

. The analytic

L

-invariant is the ratio of the leading term of the

p

-adic

L

-function at

s = 0

to its classical counterpart,

\begin{matrix} (7.7) & L_{a n} = - \frac{L_{p}^{'} (χ ω, 0)}{L (χ, 0)} \end{matrix}

The algebraic

L

-invariant is the ratio of the

p

-adic logarithm and valuation of the

χ^{- 1}

component of the Brumer-Stark unit,

\begin{matrix} (7.8) & L_{alg} = \frac{\log_{p} {Norm}_{H_{β} / Q_{p}} (u_{p}^{χ^{- 1}})}{{ord}_{β} (u_{p}^{χ^{- 1}})} \end{matrix}

There is no difficulty in defining the ratios (7.7) and (7.8), since the quantities live in a

p

-adic field and the denominators are nonzero. The analogue of this situation for Gross's Conjecture 4.4 is more delicate. The role of the

p

-adic

L

-function is played by the Stickelberger element

Θ_{L} := Θ_{S^{'}, T} (L / F, 0) \in Z [g]

, and the analogue of the derivative at 0 is played by the image of

Θ_{L}

I / I^{2}

. The role of the classical

L

-function is played by the element

Θ_{H} := Θ_{S, T} (H / F, 0) \in Z [G]

. It is therefore not clear how to take the "ratio" of these quantities. Similarly, the role of the

p

-adic logarithm is played by

{rec}_{G} (u_{p}) \in I / I^{2}

and the role of the

p

-adic valuation is played by

{ord}_{G} (u_{p}) \in Z [G]

For this reason, we introduce in [18] an

R

-algebra

R_{L}

that is generated by an element

L

that plays the role of the analytic

L

-invariant, i.e., the "ratio" between

Θ_{L}

and

Θ_{H}

. We define

\begin{matrix} (7.9) & R_{L} = R [L] / (Θ_{H} L - Θ_{L}, L I, L^{2}, I^{2}) \end{matrix}

A key nontrivial result is that this ring, in which we have adjoined a ratio

L

between

Θ_{L}

and

Θ_{H}

, is still large enough to see

R / I^{2}

Theorem 7.7 ([18, THEOREM 3.4]). The kernel of the structure map

R \to R_{L}

I^{2}

It follows from this theorem that Gross's Conjecture is equivalent to the equality

\begin{matrix} (7.10) & {rec}_{G} (u_{p}) = L {ord}_{G} (u_{p}) in R_{L} \end{matrix}

since the right-hand side is by definition

L Θ_{H} = Θ_{L}

To prove (7.10), we define a generalized Ritter-Weiss module

\nabla_{L}

over the ring

R_{L}

that can be viewed as a gluing of the modules

\nabla_{S}^{T} (H)

and

\nabla_{S^{'}}^{T} (L)

. We show in [18, THEOREM 4.6] that the Fitting ideal Fitt

R_{L} (\nabla_{L})

is generated by the element

{rec}_{G} (u_{p}) - L {ord}_{G} (u_{p}) \in I / I^{2}

and hence that (7.10) is equivalent to

\begin{matrix} (7.1) & {Fitt}_{R_{L}} (\nabla_{L}) = 0 \end{matrix}

(For the sake of accuracy, we remark that in reality we do all of this with

(S, T)

replaced by the pair

(Σ, Σ^{'})

defined in Section 5.3, as in Section 6.)

The vanishing of Fitt

R_{L} (\nabla_{L})

is proven following the methods of Section 6. We interpret surjective homomorphisms from

\nabla_{L}

R_{L}

-modules

M

in terms of Galois cohomology classes satisfying certain local conditions. We construct a suitable Galois cohomology class valued in a module

M

using an explicit construction with group-ring valued Hilbert modular forms and their associated Galois representations. The module

M

is shown to be large enough that its Fitting ideal over

R_{L}

vanishes, whence the same is true for

\nabla_{L}

since it has

M

as a quotient.

7.5. The method of Darmon-Pozzi-Vonk

We conclude by describing a proof of Conjecture 7.4 in the case that

F

is a real quadratic field in the beautiful work of Darmon, Pozzi, and Vonk [14]. Their method is purely

p

-adic (i.e., "vertical"), rather than involving the introduction of auxiliary primes (i.e., "horizontal"). The strategy follows a rich history of arithmetic formulas proven by exhibiting both sides of an equation as certain Fourier coefficients in an equality of modular forms. For instance, Katz gave an elegant proof of Leopoldt's evaluation of the Kubota-Leopoldt

p

-adic

L

-function at

s = 1

by exhibiting an equality of

p

-adic modular forms, one of whose constant terms is the

p

-adic

L

-value and the other is the

p

-adic logarithm of a unit (see [33, $10.2]). The proof of Darmon-Pozzi-Vonk follows a similar strategy.

Let

F

be a real quadratic field,

p

an odd prime, and

H

a narrow ring class field extension of

F

(so, in particular,

p O_{F}

splits completely in

H

). Darmon-Pozzi-Vonk demonstrate an equality of certain classical modular forms of weight 2 on

Γ_{0} (p) \subset {S L}_{2} (Z)

that we denote

f_{1}

and

f_{2}

This first of these forms

f_{1}

is obtained by considering a Hida family of Hilbert modular cusp forms for

F

specializing in weight 1 to a

p

-stabilized Eisenstein series. The constant term of this weight 1 Eisenstein series vanishes because of the trivial zero of the corresponding

p

-adic

L

-function. Pozzi has described explicitly the Fourier coefficients of the derivative of this family with respect to the weight variables [39]. The key idea of DarmonPozzi-Vonk is to restrict the derivative in the antiparallel direction along the diagonal and take the ordinary projection to obtain a classical modular form of weight 2 for

Γ_{0} (p)

. The idea of taking the derivative of a family of modular forms at a point of vanishing and applying a "holomorphic projection" operator has its roots in the seminal work of Gross-Zagier [30], and appears more recently in Kudla's program for incoherent Eisenstein series [34].

Pozzi's work relates the

p

th Fourier coefficient of this diagonal restriction to the

p

adic logarithm of the Brumer-Stark unit

σ_{b} (u_{p})

for the extension

H

. To obtain the desired weight 2 form

f_{1}

Γ_{0} (p)

, one must take a certain linear combination with the diagonal restrictions of the two ordinary families of Eisenstein series passing through this weight 1 point.

The second form

f_{2}

is defined as a generating series attached to a certain rigid analytic theta cocycle. These are classes in

H^{1} ({S L}_{2} (Z [1 / p]), A^{*} / C_{p}^{*})

, where

A^{*}

denotes the group of rigid analytic nonvanishing functions on the

p

-adic upper half plane. DarmonPozzi-Vonk construct classes in this space explicitly, and study their image under the logarithmic annular residue map

H^{1} ({S L}_{2} (Z [1 / p]), A^{*} / C_{p}^{*}) \to H^{1} (Γ_{0} (p), Z_{p})

They compute the spectral expansion of the form

f_{2}

and thereby show that its nonconstant Fourier coefficients are equal to those of

f_{1}

. Meanwhile, the constant coefficient is equal to the

p

-adic logarithm of

u_{p} (b)^{a n}

. The equality of the non-constant coefficients implies that

f_{1} = f_{2}

, and hence that the constant coefficients are equal as well, i.e.,

\log_{p} (σ_{b} (u_{p})) = \log_{p} (u_{p} (b)^{a n})

as desired. It is a tantalizing problem to generalize this strategy to arbitrary totally real fields.

ACKNOWLEDGMENTS

We would like to thank the many mathematicians whose work has been highly influential in the development of the perspective that we have described here. Our work is the continuation of a long line of research connecting

L

-functions, modular forms, Galois representations, and Fitting ideals of class groups. In particular, we would like to thank Armand Brumer, David Burns, John Coates, Pierre Charollois, Pierre Colmez, Henri Darmon, Cornelius Greither, Benedict Gross, Masato Kurihara, Cristian Popescu, Alice Pozzi, Kenneth Ribet, Jurgen Ritter, Karl Rubin, Takamichi Sano, Michael Spiess, John Tate, Jan Vonk, Alfred Weiss, and Andrew Wiles.

FUNDING

The first author is supported by a grant from the National Science Foundation, DMS1901939. The second author is supported by DST-SERB grant SB/SJF/2020-21/11, SERB SUPRA grant SPR/2019/000422 and SERB MATRICS grant MTR/2020/000215.

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SAMIT DASGUPTA

Duke University, Department of Mathematics, Campus Box 90320, Durham, NC 277080320, USA, dasgupta@ math.duke.edu

MAHESH KAKDE

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India, maheshkakde@iisc.ac.in

ARITHMETIC AND DYNAMICS ON VARIETIES OF MARKOFF TYPE

ALEXANDER GAMBURD

ABSTRACT

The Markoff equation

x^{2} + y^{2} + z^{2} = 3 x y z

, which arose in his spectacular thesis in 1879 , is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we discuss Hasse principle, asymptotics of integer points, and, in particular, recent progress towards establishing forms of strong approximation on varieties of Markoff type, as well as ensuing implications, diophantine and dynamical.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14G12; Secondary 11N36, 11D45, 37P55

KEYWORDS

Markoff triples, strong approximation, nonlinear dynamics

Important though the general concepts and propositions may be with which the modern industrious passion for axiomatizing and generalizing has presented us, in algebra perhaps more than anywhere else, nevertheless I am convinced that the special problems in all their complexity constitute the stock and core of mathematics; and to master their difficulties requires on the whole the harder labor.

Hermann Weyl, The Classical Groups, 1939

1. INTRODUCTION

1.1. Andrei Andreevich Markov is one of the towering peaks of the illustrious Saint Petersburg school of number theory, alongside with Chebyshev and Linnik. A singular characteristic of this school is a deep, often subterranean, interaction between arithmetic/combinatorics and probability/dynamics. While Markov is perhaps most widely known today for the chains named after him, it is in the context of his arguably deepest work on the minima of binary quadratic forms and badly approximable numbers

^{1}

that the following equation, now bearing his name, was born:

\begin{matrix} (1.1) & x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 3 x_{1} x_{2} x_{3} \end{matrix}

describing a Markoff surface

X \subset A^{3}

. Markoff triples

M

are the solutions of (1.1) with positive integral coordinates. Markoff numbers

M \subset N

are obtained as coordinates of elements of

M

. The Markoff sequence

M^{s}

is the set of largest coordinates of an

m \in M

counted with multiplicity; the uniqueness conjecture of Frobenius [62] asserts that

M = M^{s}

All elements of

M

are gotten from the root solution

r = (1, 1, 1)

by repeated applications of an element in a set

S

, consisting of

σ \in Σ_{3}

, the permutations of the coordinates of

(x_{1}, x_{2}, x_{3})

, and of the Vieta involutions

R_{1}, R_{2}, R_{3}

A^{3}

, with

R_{1} (x_{1}, x_{2}, x_{3}) = (3 x_{2} x_{3} -

x_{1}, x_{2}, x_{3}

) and

R_{2}, R_{3}

defined similarly. Denoting by

Γ

the nonlinear group of affine morphisms of

A^{3}

generated by

S

, the set of Markoff triples

M

can be identified with the orbit of the root

r

under the action of

Γ

, that is to say,

M = Γ \cdot r

, giving rise to the Markoff tree [8]:

\begin{aligned} (13, 1, 34) < \begin{matrix} (34, 1, 89) < \dots \\ (13, 34, 1325) < \dots \end{matrix} \\ (1, 1, 1) - (1, 1, 2) - (2, 1, 5) ⟨ \\ \begin{aligned} (5, 13, 194) < \begin{matrix} (194, 13, 7561) < \dots \\ (5, 194, 2897) < \dots \end{matrix} \\ (433, 5, 6466) < \dots \\ (29, 5, 433) < \begin{matrix} (29, 433, 37666) < \dots \end{matrix} \\ (169, 29, 14701) < \dots \\ (2, 169, 985) < \dots \end{aligned} \end{aligned}

1 This work of Markoff and some of the subsequent appearances of his equation in a tremendous variety of different contexts are briefly discussed in Section 2.

The first few members of

M

are

1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, \dots

The sequence

M^{s}

is sparse, as shown by Zagier [147]:

\begin{matrix} (1.2) & \sum_{\begin{matrix} m \in M^{s} \\ m \leq T \end{matrix}} 1 \sim c (\log T)^{2} as T \to \infty (c > 0) \end{matrix}

1.2. The origins of investigations which underlie "the stock and core" of this report date back to August of 2005 and involve a "special problem" pertaining to Markoff numbers; here is Peter Sarnak's recollection [126]: "For me the starting point of this investigation was in 2005 when Michel and Venkatesh asked me about the existence of poorly distributed closed geodesics on the modular surface. It was clear that Markov's constructions of his geodesics using his Markov equation provided what they wanted but they preferred quadratic forms with square free discriminant. This raised the question of sieving in this context of an orbit of a group of (nonlinear) morphisms of affine space. The kind of issues that one quickly faces in attempting to execute such a sieve are questions of the image of the orbit when reduced

mod q

and interestingly whether certain graphs associated with these orbits are expander families.

^{2}

Gamburd in his thesis had established the expander property in some simpler but similar settings and he and I began a lengthy investigation into this sieving problem in the simpler setting when the group of affine morphisms acts linearly (or what we call now the affine linear sieve)."

The question posed by Michel and Venkatesh arose in the course of their joint work with Einsiedler and Lindenstrauss [58,59] on generalizations of Duke's theorem [57]; formulated in terms of Markoff numbers, it leads to the following:

Conjecture 1. There are infinitely many square-free Markoff numbers.

As detailed in [21], an application of sieve methods in the setting of affine orbits leads to and demands an affirmative answer to the question as to whether Markoff graphs, obtained as a modular reduction of the Markoff tree,

^{3}

form a family of expanders. Numerical experiments by de Courcy-Ireland and Lee [55], as well as results detailed in Section 2.5, are compelling in favor of the following superstrong approximation conjecture for Markoff graphs:

Conjecture 2. The family of Markoff graphs

X^{*} (p)

forms a family of expanders.

Before attacking this conjecture, asserting high connectivity of Markoff graphs, one has to confront the question of their connectivity, that is to say, the issue of the strong approximation for Markoff graphs:

FIGURE 1

Markoff graph mod 7. In [54] it is proved that the Markoff graphs are not planar for primes greater than 7.

Conjecture 3. The map

π_{p} : M \to X^{*} (p)

is onto, that is to say, Markoff graphs

X^{*} (p)

are connected.

While Conjectures 1 and 2 have withstood our protracted attack over the past 17 years, much progress has been made on parallel questions in the case of affine linear maps. We will return to the recent resolution of Conjecture 3, and resulting progress on diophantine properties of Markoff numbers in Section 1.5.

1.3. Before describing the general setting of Affine Linear Sieve, it is instructive to briefly examine an example which is in many ways parallel to the Markoff situation, namely integral Apollonian packings [63,127]. A theorem of Descartes asserts that

x_{1}, x_{2}, x_{3}, x_{4} \in R^{4}

are the curvatures of four mutually tangent circles in the plane if

\begin{matrix} (1.3) & 2 (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}) = {(x_{1} + x_{2} + x_{3} + x_{4})}^{2} \end{matrix}

Given an initial configuration of 4 such circles, we fill in repeatedly the lune regions with the unique circle which is tangent to 3 sides (which is possible by a theorem of Apollonius). In this way we get a packing of the outside circle by circles giving an Apollonian packing. The interesting diophantine feature is that if the initial curvatures are integral then so are the curvatures of the entire packing.

The numbers in the circles in Figure 2 indicate their curvatures; note that by convention the outer circle has negative curvature. Viewing equation (1.3) as a quadratic equation in

x_{1}

, we see that the two solutions are related as

x_{1} + x_{1}^{'} = 2 x_{2} + 2 x_{3} + 2 x_{4}

, the crucial point being that the Vieta involutions in this case are given by linear maps

A_{1}, A_{2}, A_{3}, A_{4}

where

A_{j} (e_{k}) = - 3 e_{k} + 2 (e_{1} + e_{2} + e_{3} + e_{4})

k = j

and

A_{j} (e_{k}) = e_{k}

k \neq j

(e_{1}, e_{2}, e_{3}, e_{4}

are the standard basis vectors). The configurations of 4 mutually tangent circles in the packing with initial configuration

a = (a_{1}, a_{2}, a_{3}, a_{4})

consist of points

x

in the orbit

O = Λ \cdot a

where

Λ = ⟨ A_{1}, A_{2}, A_{3}, A_{4} ⟩

is the Apollonian group. The elements

A_{j}

FIGURE 2

Integral Apollonian packing

(- 11, 21, 24, 28)

preserve

F

given by

F (x_{1}, x_{2}, x_{3}, x_{4}) = 2 (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}) - {(x_{1} + x_{2} + x_{3}, + x_{4})}^{2}

and hence

Λ \leq O_{F} (Z)

. The group

Λ

is Zariski dense in

O_{F}

, but it is thin in

O_{F} (Z)

. For example,

| {γ \in O_{F} (Z) : ∥ γ ∥ \leq T} | \sim c_{1} T^{2}

T \to \infty

, while

| {γ \in Λ : ∥ γ ∥ \leq T} | \sim c_{1} T^{δ}

, where

^{4} δ = 1.3 \dots

is the Hausdorff dimension of the limit set of

Λ

The general setting of Affine Linear Sieve, introduced in [20,21], is as follows. For

j = 1, 2, \dots, k

, let

A_{j}

be invertible integer coefficient polynomial maps from

Z^{n}

Z^{n}

(here

n \geq 1

and the inverses of

A_{j}

's are assumed to be of the same type). Let

Λ

be the group generated by

A_{1}, \dots, A_{k}

and let

O = Λ b

be the orbit of some

b \in Z^{n}

under

Λ

. Given a polynomial

f \in Q [x_{1}, \dots, x_{n}]

which is integral on

O

, the aim is to show that there are many points

x \in O

at which

f (x)

has few or even the least possible number of prime factors, in particular that such points are Zariski dense in the Zariski closure,

Zcl (O)

O

. Let

O (f, r)

denote the set of

x \in O

for which

f (x)

has at most

r

prime factors. As

r \to \infty

, the sets

O (f, r)

increase and potentially at some point become Zariski dense. Define the saturation number

r_{0} (O, f)

to be the least integer

r

such that

Zcl (O (f, r)) = Zcl (O)

. It is by no means obvious that it is finite or even if one should expect it to be so, in general. If it is finite, we say that the pair

(O, f)

saturates. In the case of linear maps, the theory by now is quite advanced and the basic result pertaining to the finiteness of the saturation number in all cases where it is expected to hold, namely in the case of the Levi factor of

G = Zcl (Λ)

being semisimple, 5 has been established [123]. Both strong and superstrong approximation, particularly for thin

4 This result can be deduced from the work of Lax and Phillips [93]. A beautiful overview of striking developments pertaining to dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings (and beyond) is contained in Hee Oh's ICM report [114].

5 On the other hand, as detailed in [21, 85, 123], when torus intervenes, the saturation most likely fails. Tori pose particularly difficult problems, in terms of sparsity of elements in an orbit, strong approximation and diophantine properties: see [104] for a discussion of Artin's Conjecture in the context of strong approximation.
groups such as the Apollonian group, are crucial ingredients in executing Brun combinatorial sieve in this setting.

1.4. The strong approximation for

{S L}_{n} (Z)

, asserting that the reduction

π_{q}

modulo

q

is onto, is a consequence of the Chinese remainder theorem; its extension to arithmetic groups is far less elementary but well understood [118]. If

S

is a finite symmetric generating set of

{SL}_{n} (Z)

, strong approximation is equivalent to the assertion that the Cayley graphs

E ({S L}_{n} (Z / q Z), π_{q} (S))

are connected. The quantification of this statement, asserting that they are in fact highly-connected, that is to say, form a family of expanders, is what we mean by superstrong approximation. The proof of the expansion property for

{S L}_{2} (Z)

has its roots in Selberg's celebrated lower bound [131] of

\frac{3}{16}

for the first eigenvalue of the Laplacian on the hyperbolic surfaces associated with congruence subgroups of

{S L}_{2} (Z)

. The generalization of the expansion property to

G (Z)

where

G

is a semisimple matrix group defined over

Q

is also known thanks to developments towards the general Ramanujan conjectures that have been established; this expansion property is also referred to as property

τ

for congruence subgroups [133].

Let

Γ

be a finitely generated subgroup of

{GL}_{n} (Z)

and let

G = Zcl (Γ)

. The discussion of the previous paragraph applies if

Γ

is of finite index in

G (Z)

. However, if

Γ

is thin, that is to say, of infinite index in

G (Z)

, then

vol (G (R) ∖ Γ) = \infty

and the techniques used to prove both of these properties do not apply. It is remarkable that, under suitable natural hypothesis, strong approximation continues to hold in this thin context, as proved by Matthews, Vaserstein, and Weisfeiler in 1984 [105,143]. That the expansion property might continue to hold for thin groups was first suggested by Lubotzky and Weiss in 1993 [101]; for

{S L}_{2} (Z)

, the issue is neatly encapsulated in the following 1-2-3 question of Lubotzky [99]. For a prime

p \geq 5

and

i = 1, 2, 3

, let us define

S_{p}^{i} = {(\begin{array}{cc} 1 & i \\ 0 & 1 \end{array}), (\begin{array}{ll} 1 & 0 \\ i & 1 \end{array})}

. Let

E_{p}^{i} = E ({S L}_{2} (Z / p Z), S_{p}^{i})

, the Cayley graph of

{S L}_{2} (Z / p Z)

with respect to

S_{p}^{i}

. By Selberg's theorem,

E_{p}^{1}

and

E_{p}^{2}

are families of expander graphs. However, the group

⟨ (\begin{array}{ll} 1 & 3 \\ 0 & 1 \end{array}), (\begin{array}{ll} 1 & 0 \\ 3 & 1 \end{array}) ⟩

has infinite index in

{S L}_{2} (Z)

and thus does not come under the purview of Selberg's theorem.

In my thesis [66], extending the work of Sarnak and Xue [129], [128] for cocompact arithmetic lattices, a generalization of Selberg's theorem for infinite index "congruence" subgroups of

{S L}_{2} (Z)

was proved; for such subgroups with a high enough Hausdorff dimension of the limit set, a spectral gap property was established. Following the groundbreaking work of Helfgott [77] (which builds crucially on sum-product estimate in

F_{p}

due to Bourgain, Katz, and Tao [27]), Bourgain and Gamburd [13] gave a complete answer to Lubotzky's question. The method introduced in

[12, 13]

and developed in a series of papers [14-19] became known as "Bourgain-Gamburd expansion machine"; thanks to a number of major developments by many people

[22, 28, 35, 82, 91, 115, 120, 122, 124]

, the general superstrong approximation for thin groups is now known. The state-of-the-art is summarized in Thin groups and superstrong approximation [36] which contains an expanded version of most of the invited lectures from the eponymous MSRI "Hot Topics" workshop, in the surveys by Breuillard [33] and Helfgott [78], and in the book by Tao "Expansion in finite simple groups of Lie type" [140].

1.5. We return to the progress on Conjecture 3 [23-26]. Our first result [25] asserts that there is a very large orbit.

Theorem 1. Fix

ε > 0

. Then for

p

large prime, there is a

Γ

orbit

C (p)

X^{*} (p)

for which

\begin{matrix} (1.4) & | X^{*} (p) ∖ ⨀ (p) | \leq p^{ε} \end{matrix}

(note that

| X^{*} (p) | \sim p^{2}

), and any

Γ

orbit

D (p)

satisfies

^{6}

\begin{matrix} (1.5) & | D (p) | ≫ (\log p)^{\frac{1}{3}} \end{matrix}

The proof, discussed in section 3, establishes the strong approximation conjecture, unless

p^{2} - 1

is a very smooth number. In particular, the set of primes for which the strong approximation conjecture fails is very small.

Theorem 2. Let

E

be the set of primes for which the strong approximation conjecture fails. For

ε > 0

, the number of primes

p \leq T

with

p \in E

is at most

T^{ε}

, for

T

large.

Very recently, in a remarkable breakthrough, using geometric techniques involving Hurwitz stacks, degeneration, and some Galois theory, William Chen [45] proved the following result:

Theorem 3. Every

Γ

orbit

D (p)

has size divisible by

p

Combining Theorems 1 and 3 establishes Conjecture 3 for all sufficiently large primes; in combination with the following result established in [26], namely

Theorem 4. Assume that

X^{*} (Z / p Z)

is connected. Then

X^{*} (Z / p^{k} Z)

is connected for all

k

. it yields

Theorem 5. For all sufficiently large primes

p

, the group

Γ

acts minimally on

X^{*} (Z_{p})

We remark that Theorem 5 is not true for

X^{*} (R)

; cf. section 4.1. While Conjecture 1 remains out of reach, the progress on strong approximation allows us to establish the following result on the diophantine

^{7}

properties of Markoff numbers [25]:

Theorem 6. Almost all Markoff numbers are composite, that is,

\sum_{\begin{matrix} p \in M^{s} \\ p prime, p \leq T \end{matrix}} 1 = o (\sum_{\begin{matrix} m \in M^{s} \\ m \leq T \end{matrix}} 1)

It is worth contrasting this result with the state of knowledge regarding the sequence

H_{n} = 2^{n} + b

, which is just a little more sparse than the sequence of Markoff numbers, for which, by Zagier's result (1.2), we have

M_{n} \sim A^{\sqrt{n}}

. Even assuming the generalized Riemann Hypothesis, which allowed Hooley [79] to give a conditional proof of Artin's conjecture (cf. footnote 5), was not sufficient to establish that almost all members of the sequence

H_{n}

are composite: the conditional proof in [80] necessitated postulating additional "Hypothesis A."

1.6. The methods of proof of Theorems 1, 2, 4 discussed in Section 3 are robust enough to enable handling their extension to more general Markoff-type cubic surfaces, namely

\begin{matrix} (1.6) & X_{k} : Φ (x_{1}, x_{2}, x_{3}) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} - x_{1} x_{2} x_{3} = k \end{matrix}

where the real dynamics was studied by Goldman [73], as discussed in Section 4.1; the family of surfaces

S_{A, B, C, D} \subset C^{3}

given by

\begin{matrix} (1.7) & x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{1} x_{2} x_{3} = A x_{1} + B x_{2} + C x_{3} + D \end{matrix}

where the real dynamics was studied by Cantat [38], as discussed in Section 4.2; those in [60] and even the general such nondegenerate cubic surface

\begin{matrix} (1.8) & Y = Y (α, β, γ, δ) : \sum_{i, j = 1}^{3} α_{i j} x_{i} x_{j} + \sum_{j = 1}^{3} β_{j} x_{j} + γ = δ x_{1} x_{2} x_{3} \end{matrix}

with

α_{i j}, β_{j}, γ, δ

being integers.

The group

Γ_{Y}

is again generated by the corresponding Vieta involutions

R_{1}, R_{2}, R_{3}

. For such a

Y

and action

Γ_{Y}

, one must first show that there are only finitely many finite orbits in

Y (\bar{Q})

, and that these may be determined effectively. The analogue of Conjecture 1 for

Y

is that for

p

large,

Γ_{Y}

has one big orbit on

Y (Z / p Z)

and that the remaining orbits, if there are any, correspond to one of the finite

\bar{Q}

orbits determined above.

The determination of the finite orbits of

Γ

X_{k} (\bar{Q})

and on

S_{A, B, C, D} (\bar{Q})

has been carried out in [56] and [96], respectively. Remarkably for these, the

Γ

action on affine 3space corresponds to the (nonlinear) monodromy group for PainlevÃ© VI equations on their parameter spaces. In this way the finite orbits in question turn out to correspond bijectively to those PainlevÃ© VI's which are algebraic functions of their independent variable.

In this setting of the more general surfaces

Y

in (1.8), strong approximation for

Y (Z_{S})

, where

S

is the set of primes dividing

α_{11}, α_{22}, α_{33}

(so that

Γ_{Y}

preserves the

S

-integers

Z_{S}

), will follow from Conjecture 1 for

Y

(and the results we can prove towards it, as in Theorem 2) once we have a point of infinite order in

Y (Z_{S})

. If there is no such point, we can increase

S

or replace

Z

O_{K}

, the ring of integers in a number field

K / Q

, to produce such a point and with it strong approximation for

Y ({(O_{K})}_{S})

Vojta's conjectures and the results proven towards them [51,141] assert that cubic and higher-degree affine surfaces typically have few

S

-integral points. In the rare cases where these points are Zariski dense, such as tori (e.g.,

N (x_{1}, x_{2}, x_{3}) = k

where

N

is the norm form of a cubic extension of

Q

), strong approximation fails. So these Markoff surfaces appear to be rather special affine cubic surfaces not only having a Zariski dense set of integral points, but also a robust strong approximation.

1.7. Zagier's result (1.2) can be viewed as a statement about asymptotic growth of integral points on the Markoff variety,

| X (Z) \cap B (T) | \sim (\log T)^{2}

. In Section 5 we discuss the work in [68], establishing an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation

\begin{matrix} (1.9) & x_{1}^{2} + x_{2}^{2} + \dots + x_{n}^{2} = a x_{1} x_{2} \dots x_{n} + k \end{matrix}

giving an interpretation of the exponent of growth, which for

n > 3

is not integral, in terms of the unique parameter for which there exists a certain conformal measure on a projective space.

1.8. The issue of the existence of a single integral solution to (1.9) for general

a

and

k

, even for

n = 3

, is quite subtle; see [112,130]. In the work of Ghosh and Sarnak [71], the Hasse principle is established to hold for Markoff-type cubic surfaces

X (k)

given by (1.6) for almost all

k

, but it also fails to hold for infinitely many

k

; this work is discussed in Section 6.

1.9. Regrettably, the space/time constraints prevented us from covering cognate results pertaining to arithmetic and dynamics on

K 3

surfaces; see

[37, 65, 106, 108, 109, 135]

and references therein. The Markoff equation over quadratic imaginary fields is studied in [134]. Potential cryptographic applications of Markoff graphs are discussed in [64].

1.10. To conclude this introduction, let us note that

X_{k}

is the relative character variety of representations of the fundamental group of a surface of genus 1 with one puncture to

{S L}_{2}

. The action of the mapping class group is that of

Γ

. More generally, the (affine) relative character variety

V_{k}

of representation of

π_{1} (Σ_{g, n})

, a surface of genus

g

with

n

punctures, into

{S L}_{2}

is defined over

Z

, and one can study the diophantine properties of

V_{k} (Z)

. In the work of Whang [144-146], it was shown that

V_{k}

has a projective compactification relative to which

V_{k}

is "log-Calabi-Yau." According to the conjectures of Vojta, this places

V_{k}

as being in the same threshold setting as affine cubic surfaces. Moreover,

V_{k} (Z)

has a full descent in that the mapping class group acts via nonlinear morphisms on

V_{k} (Z)

with finitely many orbits. These and more general character varieties connected with higher TeichmÃ¼ller theory offer a rich family of threshold affine varieties for which one can approach the study of integral points.

2. THE UNREASONABLE(?) UBIQUITY OF MARKOFF EQUATION

Markoff equation and numbers appear in a surprising variety of contexts: see, for example, [1] (subtitled Mathematical Journey from Irrational Numbers to Perfect Matchings) and the references therein.

2.1. The Markoff chain. Equation (1.1) was discovered by Markoff in 1879 in his work on badly approximable numbers. As the sentiment

^{8}

expressed by Frobenius [62] in 1913 seems to remain true today, we briefly review the context and statement of Markoff's theorem.

Let

α

be an irrational number. A celebrated theorem of Hurwitz asserts that

α

admits

infinitely many rational approximations

p / q

such that

| α - \frac{p}{q} | < \frac{1}{\sqrt{5} q^{2}}

, and, moreover, that if

α

{G L}_{2} (Z)

-equivalent to the Golden Ratio

θ_{1} = (1 + \sqrt{5}) / 2

, in the sense that

α = \frac{a θ_{1} + b}{c θ_{1} + d}

for some integers

a, b, c, d

with

a d - b c = \pm 1

, the above result is sharp and the constant

\frac{1}{\sqrt{5}}

cannot be replaced by any smaller.

Suppose next that

α

is not

{G L}_{2} (Z)

-equivalent to

θ_{1}

. Then the result of Markoff's doctoral advisors, Korkine and Zolotareff, [88] asserts that

α

admits infinitely many rational approximations

p / q

such that

| α - \frac{p}{q} | < \frac{1}{\sqrt{8} q^{2}}

, and, moreover, that the constant

\frac{1}{\sqrt{8}}

is sharp if and only if

α

{G L}_{2} (Z)

-equivalent

θ_{2} = 1 + \sqrt{2}

The general result found by Markoff in his Habilitation and published in 1879 and 1880 in Mathematische Annalen is as follows.

Markoff's Theorem. Let

M = {1, 2, 5, 13, 29, 34, 89, 169, 194, \dots}

be the sequence of Markoff numbers. There is a sequence of associated quadratic irrationals

θ_{i} \in Q (\sqrt{Δ_{i}})

, where

Δ_{i} = 9 m_{i}^{2} - 4

and

m_{i}

is the

i

th element of the sequence, with the following property. Let

α

be a real irrational, not

G L_{2} (Z)

-equivalent to any of the numbers

θ_{i}

whenever

m_{i} <

m_{j}

. Then

α

admits infinitely many rational approximations

p / q

with

| α - \frac{p}{q} | < \frac{m_{j}}{\sqrt{Δ_{j}} q^{2}}

; the constant

m_{j} / \sqrt{Δ_{j}}

is sharp if and only if

α

G L_{2} (Z)

-equivalent to

θ_{h}

, for some

h

such that

m_{h} = m_{j}

2.2. Continued fractions and binary quadratic forms. The first paper by Markoff [102] used the theory of continued fractions, while the second memoir [103] was based on the theory of binary indefinite quadratic forms, with the final result stated as a theorem on minima of binary indefinite quadratic forms.

The alternative approach based on indefinite binary quadratic forms was the subject of an important memoir by Frobenius [62] and complete details were finally provided by Remak [121] and much simplified by Cassels [39,40].

2.3. The geometry of Markoff numbers. A third way of looking at the problem, via hyperbolic geometry, was introduced by Gorshkov [74] in his thesis of 1953, but published only in 1977. The connection with hyperbolic geometry was rediscovered, in a somewhat different way, by Cohn [46]. The paper by Caroline Series [132] contains a beautiful exposition of the problem in this context.

2.4. Cohn tree and Nielsen transformations. Cohn is also credited for the interpretation of the problem [47] in the context of primitive words in

F_{2}

, the free group on two generators. Its automorphism group

Φ_{2} = Aut (F_{2})

is generated by the following Nielsen transformations:

(a, b)^{P} = (b, a), (a, b)^{σ} = (a^{- 1}, b), (a, b)^{U} = (a^{- 1}, a b)

. Let

V = σ U

. Then

(a, b)^{V} = (a, a b)

The Cohn tree is a binary tree with root

a b

, branching to the top with

U

and to the bottom with

V

Markoff numbers are obtained from the Cohn tree by taking a third of the trace of the matrix obtained by substituting the matrices

A = (\begin{array}{ll} 5 & 2 \\ 2 & 1 \end{array})

and

B = (\begin{array}{ll} 2 & 1 \\ 1 & 1 \end{array})

in place of

a, b

and performing the matrix multiplication.

2.5. Nielsen systems and product replacement graphs. Conjecture 3 is a special case of Conjecture

Q

made by McCullough and Wanderley [107] in the context of Nielsen systems and product replacement graphs.

Given a group

G

, the product replacement graph

Γ_{k} (G)

introduced in [42] in connection with computing in finite groups is defined as follows. The vertices of

Γ_{k} (G)

consist of all

k

-tuples of generators

(g_{1}, \dots, g_{k})

of the group

G

. For every

(i, j), 1 \leq i

j \leq k, i \neq j

, there is an edge corresponding to transformations

L_{i, j}^{\pm}

and

R_{i, j}^{\pm}

, where

R_{i, j}^{\pm} : (g_{1}, \dots, g_{i}, \dots, g_{k}) \to (g_{1}, \dots, g_{i} \cdot g_{j}^{\pm 1}, \dots, g_{k})

and

L_{i, j}^{\pm}

defined similarly. The graphs

Γ_{k} (G)

are regular, of degree

4 k (k - 1)

, possibly with loops and multiple edges. The connectivity of

Γ_{k} (G)

has been the subject of intensive recent investigations; for

G = {S L}_{2} (p)

and

k \geq 3

, it was established by Gilman in [72].

In the case of the free group

F_{k}

, the moves

L_{i, j}^{\pm}

and

R_{i, j}^{\pm}

defined above correspond to Nielsen moves on

Γ_{k} (F_{k})

. For every group

G

, the set

Γ_{k} (G)

can be identified with

E = Epi (F_{k}, G)

, the set of epimorphisms from

F_{k}

onto

G

, and the group

A = Aut (F_{k})

acts on

E

in the following way: if

α \in A

and

φ \in E, α (φ) = φ \cdot α^{- 1}

. A long-standing problem is whether

Aut (F_{k})

has property (T) for

k \geq 4

; in [100] Lubotzky and Pak observed that a positive answer to this problem implies the expansion of

Γ_{k} (G)

for all

G

and proved that

Γ_{k} (G)

are expanders when

G

is nilpotent of class

l

and both

k

and

l

are fixed. Property (T) for

Aut (F_{k})

for

k \geq 5

was recently established in [84].

^{9}

Note that

Aut (F_{2})

and

Aut (F_{3})

do not satisfy property (T), while the problem is still open for

k = 4

In a joint work with Pak [69], we established a connection between the expansion coefficient of the product replacement graph

Γ_{k} (G)

and the minimal expansion coefficient of a Cayley graph of

G

with

k

generators, and, in particular, proved that for

k > 3

the product

9 The proof stems from the groundbreaking observation by Ozawa [116] that

G

satisfies Kazhdan's property (T) if there exist

λ > 0

and finitely many elements

ξ_{i}

R [G]

such that

Δ^{2} - λ Δ = \sum_{i} ξ_{i}^{*} ξ_{i}

where

Δ

is the Laplacian of the finite symmetric generating set of

G

.
replacement graphs

Γ_{k} (SL (2, p))

form an expander family under assumption of strong uniform expansion of SL

(2, p)

k

generators. In a joint work with Breuillard [34], combining the "expansion machine" [13] with the uniform Tits Alternative

^{10}

established by Breuillard [32], we proved that Cayley graphs of

SL (2, p)

are strongly uniformly expanding for infinitely many primes of density one. Consequently, the following form of nonlinear superstrong approximation is obtained:

Theorem 7. Let

k > 3

. The family of product replacement graphs

{Γ_{k} (SL (2, p_{n}))}_{n}

forms a family of expanders for infinitely many primes

p_{n}

of density one.

As detailed in [107], the situation is different for the product replacement graph of

SL (2, F_{p})

on 2 generators, due to Fricke identity for

2 \times 2

matrices

M

and

N

\begin{matrix} (2.1) & tr (M)^{2} + tr (N)^{2} + tr (M N)^{2} = tr (M) tr (N) tr (M N) + tr ([M, N]) + 2 \end{matrix}

Letting

x_{1} = tr (M), x_{2} = tr (N), x_{3} = tr (M N)

, the

Q

conjecture

^{11}

in [107] amounts to the assertion of the strong approximation for the surfaces

\begin{aligned} X_{k} : Φ (x_{1}, x_{2}, x_{3}) & = k \\ (2.3) & Φ (x_{1}, x_{2}, x_{3}) & = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} - x_{1} x_{2} x_{3} \end{aligned}

and

k = tr ([M, N]) + 2

, with Markoff surface

^{12}

being the special case corresponding to

tr ([M, N]) = - 2

3. STRONG APPROXIMATION

We give a brief overview of the methods and tools used in the proof of Theorems 1 and 2 and some comments about their extensions to the setting of more general surfaces of Markoff type. Theorem 1, in the weaker form that

| C (p) | \sim | X^{*} (p) |

p \to \infty

, can be viewed as the finite field analogue of [73] where it is shown that the action of

Γ

on the compact real components of the relative character variety of the mapping class group of the once punctured torus is ergodic. As in [73] our proof makes use of the rotations

τ_{i j} \circ R_{j}

i \neq j

, where

τ_{i j}

permutes

x_{i}

and

x_{j}

. These preserve the conic sections gotten by intersecting

X^{*} (p)

with the planes

y_{k} = x_{k}

(

k

different from

i

and

j

). If

τ_{i j} \circ R_{j}

has order

t_{1}

(here

t_{1} ∣ p (p - 1) (p + 1))

, then

x

and these

t_{1}

points of the conic section are connected (i.e., are in the same

Γ

orbit). If

t_{1}

is maximal (i.e., is

p, p - 1

, or

p + 1

), then this entire conic section is connected and such conic sections in different planes which intersect are also connected. This leads to a large component which we denote by

と (p)

10 This states that if the subgroup of

{G L}_{d} (K)

(where

K

is an algebraic number field) generated by

F

is not virtually solvable, then there is some

N \in N

, depending only on

d

, such that

{(F \cup F^{- 1} \cup {1})}^{N}

contains two elements that generate a nonabelian free group.

11 See the paper of Will Chen [45] for the discussion of the relation between this conjecture and the connectivity properties of the moduli spaces of elliptic curves with

G = SL (2, p)

structures.

12 Note that the congruence

x^{2} + y^{2} + z^{2} \equiv x y z (mod 3)

has no nontrivial solutions.

If our starting rotation has order

t_{1}

which is not maximal, then the idea is to ensure that among the

t_{1}

points to which it is connected, at least one has a corresponding rotation of order

t_{2} > t_{1}

, and then to repeat. To ensure that one can progress in this way, a critical equation over

F_{p}

intervenes:

\begin{matrix} (3.1) & {\begin{cases} x + \frac{b}{x} = y + \frac{1}{y}, b \neq 1 \\ x \in H_{1}, y \in H_{2} with H_{1}, H_{2} subgroups of F_{p}^{*} (or F_{p^{2}}^{*}) \end{cases} \end{matrix}

t_{1} = | H_{1} | \geq p^{1 / 2 + δ}

(with

δ

small and fixed), one can apply the proven Riemann Hypothesis for curves over finite fields [142] to count the number of solutions to (3.1). Together with a simple inclusion/exclusion argument, this shows that one of the

t_{1}

points connected to our starting

x

has a corresponding maximal rotation and hence

x

is connected to

ℓ (p)

| H_{1} | \leq p^{1 / 2 + δ}

then

R H

for these curves is of little use (their genus is too large), and we have to proceed using other methods. We assume that

| H_{1} | \geq | H_{2} |

so that the trivial upper bound for the number of solutions to (3.1) is

2 | H_{2} |

. What we need is a power saving in this upper bound in the case that

| H_{2} |

is close to

| H_{1} |

, that is, a bound of the form

C_{τ} {| H_{1} |}^{τ}

, with

τ < 1, C_{τ} < \infty

(both fixed).

In the prime modulus case, there are several ways to proceed. The first and second methods are related to "elementary" proofs of the Riemann Hypothesis for curves. One can use auxiliary polynomials as in Stepanov's proof [137] of the Riemann Hypothesis for curves to give the desired power saving with an explicit

τ

(cf. [76] which deals with

x + y = 1

and

| H_{1} | = | H_{2} |

in (3.1)). The second method, giving the best upper bound, namely

20 max {{(| H_{1} | . | H_{2} |)}^{1 / 3}, \frac{| H_{1} | . | H_{2} |}{p}}

, is due to Corvaja and Zannier [53]. It uses their method for estimating the greatest common divisor of

u - 1

and

v - 1

in terms of the degrees of

u

and

v

and their supports, as well as (hyper) Wronskians.

The third method is based on SzemerÃ©di-Trotter theorem for modular hyperbolas [11], whose proof uses crucially expansion and

L^{2}

-flattening lemma in

{S L}_{2} (Z / p Z)

[16].

Theorem 8. Let

Φ : F_{p} \to {Mat}_{2} (F_{p})

be such that det

Φ

does not vanish identically and

Im Φ \cap P G L_{2} (F_{p})

is not contained in a set of the form

F_{p}^{*} \cdot g H

for some

g \in {S L}_{2} (F_{p})

and

H

a proper subgroup of

{S L}_{2} (F_{p})

. Then the following holds:

Given

ε > 0, r > 1

, there is

δ > 0

such that if

A \subset P^{1} (F_{p})

and

L \subset F_{p}

satisfy

\begin{aligned} (3.2) & 1 ≪ | A | & < p^{1 - ε} \\ (3.3) & \log | A | & < r \log | L | \end{aligned}

then

\begin{matrix} (3.4) & | {(x, y, t) \in A \times A \times L; y = τ_{Φ (t)} (x)} | < | A |^{1 - δ} | L | \end{matrix}

where for

g = (\begin{array}{ll} a & b \\ c & d \end{array}), τ_{g} (x) = \frac{a x + b}{c x + d}

While producing poor exponents

τ

, this method is robust and works in the generality that the superstrong approximation for

{S L}_{2} (Z / q Z)

has been established; in particular,
the analogue of Theorem 8 for

Z / p^{n} Z

, which follows from expansion in

{S L}_{2} (Z / p^{n} Z)

, established

^{13}

in [16], plays crucial role in the proof of Theorem 4 in [26].

The above leads to a proof of part 1 of Theorem 1. To continue, one needs to deal with

t_{1}

which is very small (here

| H_{1} | = t_{1}

which divides

p^{2} - 1

To handle these, we lift to characteristic zero and examine the finite orbits of

Γ

X (\bar{Q})

. In fact, by the Chebotarev Density Theorem, a necessary condition for Conjecture 3 to hold is that there are no such orbits other than

{0}

. Again using the rotations in the conic sections by planes, one finds that any such finite orbit must be among the solutions with

t_{j}

's roots of unity to

\begin{matrix} (3.5) & {(t_{1} + t_{1}^{- 1})}^{2} + {(t_{2} + t_{2}^{- 1})}^{2} + {(t_{3} + t_{3}^{- 1})}^{2} = (t_{1} + t_{1}^{- 1}) (t_{2} + t_{2}^{- 1}) (t_{3} + t_{3}^{- 1}) \end{matrix}

For this particular surface

X

, one can show using the inequality between the geometric and arithmetic means, that (3.5) has no nontrivial solutions for complex numbers with

| t_{j} | = 1

. For the more general surfaces

X_{k}, S_{A, B, C, D}

, and those in (1.8), there is a variety of solutions with

| t_{j} | = 1

. However, Lang's

G_{m}

Conjecture which is established effectively (see

[2, 92]

) yields that there are only finitely many solutions to these equations in roots of unity. This allows for an explicit determination of the finite orbits of

Γ_{Y}

Y (\bar{Q})

(as noted earlier for the cubic surfaces

S_{A, B, C, D}

, the long list of these orbits [96] correspond to the algebraic Painleve VI's). This

\bar{Q}

analysis leads to part 2 of Theorem 1 and, combined with the discussion above, it yields a proof of Conjecture 3, at least if

p^{2} - 1

is not very smooth. To prove Theorem 2, we need to show that there are very few primes for which the above arguments fail. This is done by extending the arguments and results in [43] and [44] concerning points

(x, y)

on irreducible curves over

F_{p}

for which ord

(x) + ord (y)

is small (here ord

(x)

is the order of

x

F_{p}^{*}

The proof of Theorem 6 in the stronger form that all Markoff numbers are highly composite, that is, for every

v \geq 1

, as

T \to \infty

\sum_{\begin{matrix} m \in M^{s}, m \leq T \\ m has at most \\ v distinct prime factors \end{matrix}} 1 = o (\sum_{\begin{matrix} m \in M^{s} \\ m \leq T \end{matrix}} 1)

makes use of counting points on

X^{*} (Z)

of height at most

T

and, in particular, Mirzakhani's orbit equidistribution [111], as well as the transitivity properties of

Γ

X^{*} (q)

for

q

a product of suitable primes

p

. The latter are provided by the results of Meiri and Puder [110]. For

p \equiv 1 (4)

for which the induced permutation action of

Γ

X^{*} (p)

is transitive, they show that the resulting permutation group is essentially the full symmetric or alternating group on

X^{*} (p)

. Applying Goursat's (disjointness) Lemma leads to the

Γ

-action on

X^{*} (p_{1} p_{2} \dots p_{k})

being transitive for any such primes

p_{1} < p_{2} < \dots < p_{k}

FIGURE 3

Level set

κ = - 2.1

FIGURE 4

Level set

κ = - 1.9

4. REAL DYNAMICS ON SURFACES OF MARKOFF TYPE

4.1. In this section we discuss the work of Goldman [73] pertaining to modular group action on real SL(2)-characters of a one-holed torus. The fundamental group

π

of the one-holed torus is the free group of rank two. The mapping class group of the 1-holed torus is isomorphic to the outer automorphism group

Out (π) ≅ GL (2, Z)

π

and acts on the moduli space of equivalence classes of

SL (2, C)

-representations of

π

; this moduli space identifies naturally with affine 3 -space

C^{3}

, using the traces of two generators of

π

and of their product as coordinates. In these coordinates, the trace of the commutator of the two generators (representing the boundary curve of the torus) is given by

κ (x, y, z) = x^{2} + y^{2} +

z^{2} - x y z - 2

, which is preserved under the action of

Out (π)

, and the action of

Out (π)

C^{3}

is commensurable with the action of the group

Γ

of polynomial automorphisms of

C^{3}

which preserve

κ

. Figures 3-8 show level sets of

κ

FIGURE 5

Level set

κ = 1.9

FIGURE 6

Level set

κ = 2.1

In [73] Goldman studied the dynamics of the

Γ

-action on the set of real points of this moduli space, and more precisely on the level sets

κ^{- 1} (t) \cap R^{3}

, for

t \in R

. The action of

Γ

preserves a Poisson structure defining a

Γ

-invariant area form on each level set. It is shown that for

t < 2

the

Γ

-action is properly discontinuous on the four contractible components of each level set and ergodic on the compact component (which is empty if

t < - 2

); the contractible components correspond to TeichmÃ¼ller spaces of complete hyperbolic structures on a one-holed torus if

t \leq - 2

, and of a torus with a single cone point singularity if

- 2 < t <

2. For

t = 2

, the level set consists of characters of reducible representations and comprises two ergodic components, for

2 < t \leq 18

the action of

Γ

on a level set is ergodic, and for

t > 18

the moduli space contains characters of discrete representations uniformizing a three-holed sphere and the action is ergodic on the complement.

4.2. The main objective of [38] is the dynamical description of elements of the mapping class group of the four-punctured sphere acting on two-dimensional slices of its

FIGURE 7

Level set

κ = 2.1

FIGURE 8

Level set

κ = 4

character variety. It also contains three striking applications of this analysis to the dynamics of the mapping class group on the character variety, to the spectrum of certain discrete SchrÃ¶dinger equations, and to PainlevÃ© sixth equation. Cantat considers the space of representations of the free group given by the presentation

F_{3} = ⟨ α, β, γ, δ ∣ α β γ δ = 1 ⟩

into

S L (2, C)

modulo conjugacy. By fixing the trace of the images of the four generators, one obtains a space that is naturally parameterized by a cubic surface

S_{A, B, C, D} \subset C^{3}

given by

x^{2} + y^{2} + z^{2} + x y z = A x + B y + C z + D

for some parameters

A, B, C, D \in C

. This surface admits three natural involutions

s_{x}, s_{y}, s_{z}

which fix two out of the three coordinates and transform the last to the other root of the quadratic. These involutions generate a group

Γ

of affine automorphisms. Automorphisms of

F_{3}

act by composition on the space of representations by preserving the trace, and the group of outer automorphisms of

F_{3}

acts on

S_{A, B, C, D}

in such a way that its image contains

Γ

as a finite index subgroup.

An element

f \in Γ

is called hyperbolic if it corresponds to a pseudo-Anosov automorphism in the mapping class group, or, equivalently, if it is not conjugated to the product of

FIGURE 9

S_{(- 0.2, - 0.2, - 0.2, 4.39)}

FIGURE 10

Projection of the stable manifold.

FIGURE 11

S_{(0, 0, 0, 3)}

FIGURE 12

Projection of the intersection of the stable manifold with the upper part of the surface.

FIGURE 13

S_{(0, 0, 0, 4.1)}

FIGURE 14

Projection of the intersection of the stable manifold with the upper part of the surface.
one or two involutions as above. Take

f \in Γ

of hyperbolic type, and compactify

S_{A, B, C, D}

by taking its closure in

P^{3}

. The divisor at infinity is a cycle of three rational curves. By conjugating

f

in a suitable way, one can make it algebraically stable in the sense of Fornaess and Sibony [61], so that it contracts all curves at infinity to a single superattracting fixed point. One can then prove that at that point the map is locally conjugated to a monomial map whose spectral radius

λ (f)

is greater than 1 . This enables one to define the Green functions

G_{\pm} f = lim_{n} λ (f)^{- n} \log^{+} | f^{\pm n} |

, and show that they are plurisubharmonic, continuous, and possess natural invariance properties. It follows that the positive measure

μ_{f} = d d^{c} G^{+} f \land d d^{c} G^{-} f

is well defined and

f

-invariant. Moreover,

μ_{f}

turns out to be mixing and the unique measure of maximal entropy equal to

\log λ (f)

. All these properties are reminiscent of the dynamics of HÃ©non mappings in the complex plane, and are proved analogously. Next, assume all coefficients

A, B, C, D

are real, and suppose the real part

S_{A, B, C, D} (R)

is connected (in which case it is homeomorphic to the sphere minus four points). The main theorem of the paper states that the support of the measure

μ_{f}

is then included in

S_{A, B, C, D} (R)

and that the induced map on

S_{A, B, C, D} (R)

is uniformly hyperbolic on its nonwandering set. The proof of this striking theorem uses deep results by Bedford and Smillie [6] on the characterization of nonhyperbolic real HÃ©non maps having the same entropy as their complexification and relies on a delicate geometrical analysis of the possibilities for the intersection of stable and unstable manifolds in

S_{A, B, C, D} (R)

5. AN ASYMPTOTIC FORMULA FOR INTEGER POINTS ON MARKOFF-HURWITZ VARIETIES

For integer parameters

n \geq 3, a \geq 1

, and

k \in Z

, consider the Diophantine equation

\begin{matrix} (5.1) & x_{1}^{2} + x_{2}^{2} + \dots + x_{n}^{2} = a x_{1} x_{2} \dots x_{n} + k \end{matrix}

We call this the generalized

^{14}

Markoff-Hurwitz equation. In this section we count solutions to (5.1) in integers, which we call Markoff-Hurwitz tuples. More precisely, let

V

be the affine subvariety of

C^{n}

cut out by (5.1). In a joint work with Magee and Ronan [68], we investigated the asymptotic size of the set

V (Z) \cap B (R)

where

B (R)

is the ball of radius

R

in the

ℓ^{\infty}

-norm on

R^{n} \subset C^{n}

. Perhaps somewhat surprisingly, the asymptotic growth for

n \geq 4

is not of the order

(\log R)^{n - 1}

, as was first noticed by Baragar [4], who subsequently in [5] proved that there is a number

β = β (n)

such that when

k = 0

, if

V (Z) - {(0, 0, \dots, 0)}

is nonempty then

\begin{matrix} (5.2) & | V (Z) \cap B (R) | = (\log R)^{β + o (1)} \end{matrix}

R \to \infty

In [5] the following bounds for the exponents

β (n)

were also obtained:

\begin{aligned} β (3) = 2 \\ (5.5) & β (4) \in (2.430, 2.477) \\ β (5) \in (2.730, 2.798) \\ β (6) \in (2.963, 3.048) \end{aligned}

and, in general,

\frac{\log (n - 1)}{\log 2} < β (n) < \frac{\log (n - 1)}{\log 2} + o (n^{- 0.58})

The following problems were posed by Silverman in 1995 [136] (in the setting of

k = 0)

Problem 1. Is there is a true asymptotic formula for

| V (Z) \cap B (R) |

with main term proportional to

\log (R)^{β}

Problem 2. Is

β (n)

irrational?

In [68] a complete answer to Problem 1 was obtained by extending Baragar's exponential rate of growth estimate to a true asymptotic formula.

^{15}

When

k > 0

, there are certain exceptional families of solutions to (5.1) that have a different quality of growth and, for fixed

k, a, n

, we write

&

for the set of exceptional tuples. We obtain the following theorem for the asymptotic number of Markoff-Hurwitz tuples:

Theorem 9. For each

(n, a, k)

with

V (Z) - E

infinite, there is a positive constant

c =

c (n, a, k)

such that

| (V (Z) - E) \cap B (R) | = c (\log R)^{β} + o ((\log R)^{β})

Here

β

is the same constant as in (5.2).

After renormalizing (5.1), which allows us to set

a = 1

, and rearranging entries, Markoff-Hurwitz transformations induce the moves

\begin{matrix} (5.4) & λ_{j} (z_{1}, \dots, z_{n}) = (z_{1}, \dots, \hat{z_{j}}, \dots, z_{n}, (\prod_{i \neq j} z_{i}) - z_{j}), 1 \leq j \leq n - 1 \end{matrix}

on ordered tuples of real numbers. Above,

\hat{∙}

denotes omission. If sufficiently many of the

z_{i}

are large, the move

λ_{j}

can be approximated by

z \mapsto (z_{1}, \dots, \hat{z_{j}}, \dots, z_{n}, \prod_{i \neq j} z_{i})

to high accuracy relative to the largest entries of

z

. When the

z_{i}

are positive, at the level of logarithms this corresponds to

(\log z_{1}, \log z_{2}, \dots, \log z_{n}) \mapsto (\log z_{1}, \dots, \hat{\log z_{j}}, \dots, \log z_{n}, \sum_{i \neq j} \log z_{i})

Thus one is naturally led to study the linear semigroup generated by linear maps

\begin{matrix} (5.5) & γ_{j} (y_{1}, y_{2}, \dots, y_{n}) = (y_{1}, \dots, \hat{y_{j}}, \dots, y_{n}, \sum_{i \neq j} y_{i}) \end{matrix}

on ordered

n

-tuples

(y_{1}, \dots, y_{n})

Let

Γ = {⟨ γ_{1}, \dots, γ_{n - 1} ⟩}_{+}

where we have written a " + " to indicate we are generating a semigroup, not a group.

An important idea in [68] that explains why we are able to make progress on the counting problem is that we replace the generators of

Γ

with the countably infinite generating set

T_{Γ} = {γ_{n - 1}^{A} γ_{j} : A \in Z_{\geq 0}, 1 \leq j \leq n - 2}

and then consider the semigroup

Γ^{'} = {⟨ T_{Γ} ⟩}_{+}

Both

Γ

and

Γ^{'}

preserve the nonnegative ordered hyperplane

\begin{matrix} (5.6) & H \equiv {(y_{1}, \dots, y_{n}) \in R_{\geq 0}^{n} : y_{1} \leq y_{2} \leq \dots \leq y_{n}, \sum_{j = 1}^{n - 1} y_{j} = y_{n}} \subset R_{\geq 0}^{n} \end{matrix}

any element of

Γ

maps ordered tuples in

R_{\geq 0}^{n}

into

H

. Therefore the study of orbits of

Γ

and

Γ^{'}

on ordered tuples boils down to the study of orbits in

H

. We can use the basis

e_{j} = (0, \dots, 0, \underset{j}{\underset{⏟}{1}}, 0, \dots, 0, 1)

for the subspace spanned by

H

. This basis clarifies the action of

Γ^{'}

When

n = 3

, the linear map

σ : H \to H

defined by

\begin{matrix} (5.7) & σ (a, b, a + b) = order (b - a, a, b) \end{matrix}

where order puts a tuple in ascending order from left to right, is such that for

j = 1, 2

we have

σ γ_{j} . y = y

for all

y \in H

. Repeatedly applying the map

σ

to a triple

(a, b, a + b)

with

a \leq b \in Z

performs the Euclidean algorithm on

a, b

. However, one application of

σ

corresponds in general to less than one step of the algorithm. Replacing

Γ

with

Γ^{'}

corresponds to speeding this up so one whole step of the Euclidean algorithm corresponds to one semigroup generator. As for counting, the orbit of

(0, 1, 1)

under

Γ

is precisely those

(a, b, a + b)

with

(a, b) = 1

and thus can be counted by elementary methods.

When

n = 3

, the semigroup

Γ^{'}

is generated by

g_{A} := γ_{2}^{A} γ_{1} = (\begin{array}{cc} 0 & 1 \\ 1 & A + 1 \end{array})

with respect to the basis

{e_{1}, e_{2}}

. These generators are classically connected with continued fractions by the formulae

(\begin{array}{cc} 0 & 1 \\ 1 & A_{1} \end{array}) (\begin{array}{cc} 0 & 1 \\ 1 & A_{2} \end{array}) \dots (\begin{array}{cc} 0 & 1 \\ 1 & A_{k} \end{array}) = (\begin{array}{cc} ⋆ & b \\ ⋆ & d \end{array}), \frac{b}{d} = \frac{1}{A_{1} + \frac{1}{A_{2} + ⋱ \cdot \frac{1}{A_{k}}}}

FIGURE 15

When

n = 4

, the semigroup elements map

Δ = H / R +

into a strictly smaller subset. After iteration, this leads to more and more empty space (see also Figure 16). This does not occur when

n = 3

, as one can also see from the picture: the action of the group elements

γ_{2}

and

γ_{3}

on the vertical coordinate axis is a copy of the

n = 3

dynamics

When

n = 4

, the semigroup

Γ

acts in the basis given by the

e_{i}

γ_{1} = (\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \end{array}), γ_{2} = (\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \end{array}), γ_{3} = (\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{array})

This semigroup appears naturally in different areas of mathematics. In most situations that this semigroup appears, as is the case in [68], the dynamics of the projective linear action of

Γ

R_{+}^{3} / R_{+}

becomes relevant. Up to the minor modification of possibly multiplying the generators on the left or right by permutation matrices, the iterated function system given by the projective linear action of

Γ

R_{+}^{3} / R_{+}

has a fractal attracting set that is known as the Rauzy gasket [95].

So the semigroups

Γ

and

Γ^{'}

are natural extensions of the Euclidean algorithm and continued fractions semigroup to higher dimensions. Writing

Δ = H / R_{+}

, we can view

Δ

as a subset of

R^{n - 2}

. The key distinction that appears when

n \geq 4

is that

Δ \neq ⋃_{j = 1}^{n - 1} γ_{j} (Δ)

and so the induced dynamics on

H / R_{+}

has "holes" as we illustrate in Figure 15.

Structure of the proof and the difficulties that arise. Here we highlight some of the main difficulties that must be overcome during the proof of Theorem 9. It is illuminating to recall the methods used by Lalley

^{16}

in [90] where the action of a Schottky subgroup

G

{S L}_{2} (R)

on the hyperbolic upper half-plane

H

is considered. Lalley obtains that, for any

x \in H

, the number

N (x, r)

of elements

γ

G

such that

d_{H} (i, γ x) - d_{H} (i, x) \leq r

where

d_{H}

is hyperbolic distance, satisfies

N (x, r) \approx C e^{δ r}

, where

δ = δ (G)

is the Hausdorff dimension of the limit set of

G

, and

C = C (G, x) > 0

. Lalley's proof incorporates at various stages the following arguments:

Shell argument. By repeated application of a "renewal equation," the quantity

N (x, r)

is related to a sum of

N (y, r^{'})

, where the sum is over

y

on a shell of radius

\approx c r

in a Cayley tree of

G

, and

r^{'}

is a translate of

r

that corrects for the passage between

x

and

y

. The purpose of this shell argument is that now, the points

y

lie close to

\partial H

Passage to the boundary. Each of the resulting

N (y, r^{'})

is compared to an analogous quantity

N^{*} (y^{*}, r^{'})

where

y^{*}

is a point in

\partial H

close to

y

. Because each

y

is close to

\partial H

, the errors incurred are acceptable.

Transfer operator techniques. Asymptotic formulas for

N^{*} (y^{*}, r^{'})

are obtained using the renewal method and spectral estimates for transfer operators. This gives asymptotic formulas for

N (y, r^{'})

. The main terms of the asymptotic formulas satisfy recursive relationships between different

y

Recombination. One finally has to recombine all the asymptotic formulas obtained for

N (y, r^{'})

to obtain an asymptotic formula for

N (x, r)

. This is done using the recursive formulas obtained in the previous step.

Trying to follow the method outlined above for this orbital counting problem, we first need a suitable replacement for

\partial H

. Our idea is to use the projectivization of the hyperplane

H

; we call this set

Δ

. We compare points in the orbit of

Λ

(generated by

λ_{j}

in (5.4) to points in

Δ

by taking logarithms of all coordinates and then projectivizing. This process does not necessarily lead to a point in

Δ

; there is an important parameter

α (z) = \prod_{j = 1}^{n - 2} z_{j}

that appears throughout the paper and measures how good the fit is. If

α (z)

is large, then one can, in analogy with Lalley's setting, think of

z

as being "close to the boundary."

For Lalley, the word length of

γ

is roughly proportional to the quantity

d_{H} (i, γ x) -

d_{H} (i, x)

with respect to which he counts. This implies, during the shell argument, that all the elements of the shell are roughly the same distance from

\partial H

. However, for us, there are arbitrarily long words in the generators of

Λ

for which

α (z)

is small. We solve this problem using "acceleration," by replacing

Λ

Λ^{'}

, and instead aim to follow Lalley's argument for orbits of

Λ^{'}

. This has the immediate benefit that we can guarantee that elements

z

of shells of radius

L

, with respect to

Λ^{'}

, have large

α (z)

, if we make

L

appropriately large.

However, the acceleration also has some costs to be paid. The first issue arising is that now

Λ^{'}

has countably many generators, so shells for word length on

Λ^{'}

are not finite. Instead of using shells, we use intersections of shells with the elements of the

Λ^{'}

-orbit whose coordinates are not too large. The second issue is that the original

Λ

-orbit breaks up

FIGURE 16

In the same setting

(n = 4)

of Figure 15 , we show in black the images of

Δ

under the action of all words of length 10 in the generators

{γ_{1}, γ_{2}, γ_{3}}

into countably many

Λ^{'}

-orbits. So we not only have to perform the recombination argument for

Λ^{'}

, but then have to perform an extra summation over the countably many

Λ^{'}

-orbits.

After setting up our shell argument appropriately, we must perform the passage to the boundary (i.e.,

Δ

). To this end, we compare orbits of

Λ^{'}

to orbits of

Γ^{'}

, where

Γ^{'}

is the linear semigroup. To get this to work, we must exploit the following "shadowing" feature of the map

\log

that takes logarithms of all entries of a vector. It says (roughly) that if

\log (z)

is within

ϵ

y \in H

, with

ϵ

on the scale of

α (z)^{- 2}

, then for all

λ \in Λ^{'}, \log (λ (z))

is within

ϵ

γ (\log (z))

, where

γ \in Γ^{'}

is matched with

λ

in a natural way.

The completion of the proof relies on spectral estimates for transfer operators associated to the projective linear action of

Γ^{'}

Δ

. There are three key issues arising here. First, to obtain the spectral estimates we need, we must establish that the action of

Γ^{'}

Δ

is uniformly contracting; it is important to note that this argument would not work if the acceleration had not been performed previously. Secondly, we need to establish that the relevant "log-Jacobian" cocycle over the dynamical system is not cohomologous to a lattice cocycle. Finally, but importantly, we must obtain spectral estimates for transfer operators acting on

C^{1} (Δ)

which is accomplished by adapting Liverani's approach to spectral esti-
mates from [97]. See section 4 of [68], and references therein, for the discussion of Gauss map and Gauss measure

[70, 89

] in this context.

The question of whether

β

is irrational (Problem 2) remains a tantalizing open question, and one may wonder whether it is even algebraic. Our methods do give some partial insight into the nature of this mysterious number in terms of the action of

Γ^{'}

H / R_{+}

Theorem 10. The number

β

is the unique parameter in

(1, \infty)

such that there exists a probability measure

ν_{β}

Δ = H / R_{+}

with the property

\int_{w \in Δ} f (w) d v_{β} (w) = \sum_{γ \in T_{Γ}} \int_{w \in Δ} f (γ \cdot w) {| {Jac}_{w} (γ) |}^{\frac{β}{n - 1}} d v_{β} (w)

for all

f \in C^{0} (Δ)

. We call

ν_{β}

a conformal measure.

Theorem 10 can be viewed as a partial analog of the connection between the exponent of growth of a finitely generated Fuchsian group and the Hausdorff dimension of its limit set as a result of Patterson-Sullivan theory [117,138,139]. In our setting, the lack of any symmetric space means the parameter

β

is not in any obvious way connected to the Hausdorff dimension of the compact

Γ^{'}

-invariant subset of

Δ

The issue of the existence of a single integral solution for general

a

and

k

is very subtle, even for

n = 3

, as discussed in the next section.

6. HASSE PRINCIPLE ON SURFACES OF MARKOFF TYPE

Little is known about the values at integers assumed by affine cubic forms

^{17} F

in three variables. For

k \neq 0

, set

\begin{matrix} (6.1) & V_{k, F} = {x = (x_{1}, x_{2}, x_{3}) : F (x) = k} \end{matrix}

The basic question is for which

k

V_{k, F} (Z) \neq \emptyset

, or, more generally, infinite or Zariski dense in

V_{k, F}

A prime example is

F = S

, the sum of three cubes,

\begin{matrix} (6.2) & S (x_{1}, x_{2}, x_{3}) = x_{1}^{3} + x_{2}^{3} + x_{3}^{3} \end{matrix}

There are obvious local congruence obstructions, namely that

V_{k, S} (Z) = \emptyset

k \equiv 4, 5 (mod 9)

, but beyond that, it is possible that the answers to all three questions are yes for all the other

k

's, which we call the admissible values (see [50,113]). It is known that strong approximation in its strongest form fails for

V_{k, S} (Z)

; the global obstruction coming from an application of cubic reciprocity [41,49,75]). Moreover, the authors of [94] and [7] show that

V_{1, S} (Z)

is Zariski dense in

V_{1, S}

In [71] Ghosh and Sarnak investigate the Markoff form

F = M

\begin{matrix} (6.3) & M (x) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} - x_{1} x_{2} x_{3} \end{matrix}

17 By an affine form

f

n

variables we mean

f \in Z [x_{1}, \dots, x_{n}]

whose leading homogeneous term

f_{0}

is nondegenerate and such that

f - k

is (absolutely) irreducible for all constants

k

FIGURE 17

Lattice points and fundamental set for

k = 3685

FIGURE 18

Closeup of fundamental set for

k = 3685

Except for the case of the Cayley cubic with

k = 4, V_{k; M} (Z)

decomposes into a finite number

h_{M} (k)

Γ

-orbits. For example, if

k = 0

, then

h_{M} (k) = 2

corresponds to the orbits of

(0, 0, 0)

and

(3, 3, 3)

. In order to study

h_{M} (k)

both theoretically and numerically, they give an explicit reduction (descent) for the action of

Γ

V_{k, M} (Z)

. For this purpose, it is convenient to remove an explicit set of special admissible

k

's, namely those for which there is a point in

V_{k, M} (Z)

with

| x_{j} | = 0, 1

or 2 . These

k

's take the form (i)

k = u^{2} + v^{2}

, (ii)

4 (k - 1) = u^{2} + 3 v^{2}

, or (iii)

k = 4 + u^{2}

. The number of these special

k

's (referred to as exceptional) with

0 \leq k \leq K

is asymptotic to

C^{'} \frac{K}{\sqrt{\log K}}

. The remaining admissible

k

's are called generic (all negative admissible

k

's are generic). For them Ghosh and Sarnak give the following elegant reduced forms:

FIGURE 19

Lattice points and fundamental set for

k = - 3691

FIGURE 20

Closeup of fundamental set for

k = - 3691

Proposition 11. (1) Let

k \geq 5

be generic and consider the compact set

F_{k}^{+} = {u \in R^{3} : 3 \leq u_{1} \leq u_{2} \leq u_{3}, u_{1}^{2} + u_{2}^{2} + u_{3}^{2} + u_{1} u_{2} u_{3} = k}

The points in

F_{k}^{+} (Z) = F_{k}^{+} \cap Z^{3}

are

Γ

-inequivalent, and any

x \in V_{k, M} (Z)

Γ

-equivalent to a unique point

u^{'} = (- u_{1}, u_{2}, u_{3})

with

u = (u_{1}, u_{2}, u_{3}) \in

F_{k}^{+} (Z)

(2) Let

k < 0

be admissible and consider the compact set

F_{k}^{-} = {u \in R^{3} : 3 \leq u_{1} \leq u_{2} \leq u_{3} \leq \frac{1}{2} u_{1} u_{2}, u_{1}^{2} + u_{2}^{2} + u_{3}^{2} - u_{1} u_{2} u_{3} = k}

The points in

F_{k}^{-} (Z) = F_{k}^{-} \cap Z^{3}

are

Γ

-inequivalent, and any

x \in V_{k, M} (Z)

Γ

-equivalent to a unique point

u = (u_{1}, u_{2}, u_{3}) \in F_{k}^{-} (Z)

Some consequences of this are as follows: As

k \to \pm \infty

, we have

h_{M} (k) ≪_{ε} | k |^{\frac{1}{3} + ε} .

This follows from the fact that when considering the values taken by the corresponding indefinite quadratic form in the

y

and

z

variables, for each fixed

x

, the units are bounded in number due to the restrictions imposed by the fundamental sets.

Let

h_{M}^{\pm} (k) = | F_{k}^{\pm} (Z) |

where

\pm = sgn (k)

, this being defined for any

k

. Then for generic

k, h_{M}^{\pm} (k) = h_{M} (k)

while otherwise

h_{M} (k) \leq h_{M}^{\pm} (k)

. We have

\begin{matrix} (6.4) & \sum_{\begin{matrix} k \neq 4 \\ | k | \leq K \end{matrix}} h_{M}^{\pm} (k) \sim C^{\pm} K (\log K)^{2} \end{matrix}

where

C^{\pm} > 0

and

K \to \infty

So on average the numbers

h_{M} (k)

are small. The explicit fundamental domains allow for the numerical computations; these indicate that

\begin{matrix} (6.5) & \sum_{\begin{matrix} 0 < k \leq K \\ k admissible \\ h_{M} (k) = 0 \end{matrix}} 1 \sim C_{0} K^{θ} \end{matrix}

with

C_{0} > 0

and

θ \approx 0.8875 \dots

The main result in [71] concerns the values assumed by

M

and the Hasse failures in (6.5):

Theorem 12. (i) There are infinitely many Hasse failures. More precisely, the number of

0 < k \leq K

and

- K \leq k < 0

for which the Hasse principle fails is at least

\sqrt{K} (\log K)^{- \frac{1}{4}}

for

K

large.

(ii) Fix

t \geq 0

. Then as

K \to \infty

# {| k | \leq K : k admissible, h_{M} (k) = 0} = o (K)

Hasse failures are produced by an obstruction via quadratic reciprocity. They come in two types: one via direct use of reciprocity and the second also incorporating the descent group. Recently Colliot-ThÃ©lÃ¨ne, Wei, and Xu [48] and, independently, Loughran and Mitankin [98] have shown that the obstruction of the first (but not the second type) can be explained in terms of integral Brauer-Manin obstruction. For example, if

k = 4 + 2 v^{2}

, with

v

having all of its prime factors congruent to

\pm 1 (mod 8)

and

v

congruent to

0, \pm 3, \pm 4 (mod 9)

, then

k

is admissible but

V_{k, M} (Z) = \emptyset

Part (ii) of the theorem is proved by comparing the number of points on

V_{k, M} (Z)

in certain tentacled regions gotten by special plane sections, with the expected number of solutions according to a product of local densities; the crucial point being that the variance of this comparison goes to zero on averaging

| k | \leq K

. As detailed in [71], this moving plane quadric method applies to more general cubic surfaces including those that do not carry morphisms.

ACKNOWLEDGMENTS

I would like to thank Jean Bourgain, Emmanuel Breuillard, Michael Magee, Igor Pak, Ryan Ronan, and Peter Sarnak. My various collaborations with them form the core of this report.

Figure 1 is courtesy of Matthew de Courcy-Ireland.

Figure 2 is courtesy of Elena Fuchs.

Figures 3-8 are courtesy of William Goldman.

Figures

9 - 14

are courtesy of Serge Cantat.

Figures 17-20 are courtesy of Amit Ghosh.

FUNDING

The author was supported, in part, by NSF award DMS-1603715.

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ALEXANDER GAMBURD

The Graduate Center, CUNY, New York, NY, USA, agamburd @ gmail.com

THE NUMBER OF RATIONAL POINTS ON A CURVE OF GENUS AT LEAST TWO

PHILIPP HABEGGER

ABSTRACT

The Mordell Conjecture states that a smooth projective curve of genus at least 2 defined over number field

F

admits only finitely many

F

-rational points. It was proved by Faltings in the 1980s and again using a different strategy by Vojta. Despite there being two different proofs of the Mordell Conjecture, many important questions regarding the set of

F

-rational points remain open. This survey concerns recent developments towards upper bounds on the number of rational points in connection with a question of Mazur.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14G05; Secondary 11G30, 11G50, 14G25, 14K15

KEYWORDS

Mordell conjecture, smooth projective curves, rational points, NÃ©ron-Tate height, Weil height, family of abelian varieties

1. INTRODUCTION

Mordell's Conjecture asserts the finiteness of the set of rational solutions

{(x, y) \in Q^{2} : P (x, y) = 0}

for certain bivariate polynomials

P \in Q [X, Y]

To make the statement and results precise, we will adopt the language of projective algebraic curves. Indeed, for the study of the zero set, we may assume that

P

is irreducible, even as a polynomial in

C [X, Y]

. Moreover, its homogenization defines a projective curve in the projective plane. The classical procedure of normalization allows us to resolve any singularities. The result is an irreducible smooth projective curve defined over

Q

. Its complex points define a compact Riemann surface of genus

g \in {0, 1, 2, \dots}

Conversely, let us assume we are presented with a smooth projective curve

C

defined over

Q

that is irreducible as a curve taken over

C

. The genus

g

C (C)

taken as a Riemann surface has important consequences for arithmetic questions on

C (Q)

. Indeed, Mordell's Conjecture, proved by Faltings [25], states that

# C (Q)

is finite if

g \geq 2

We begin by formulating the Mordell Conjecture in slightly higher generality. We then discuss the history of results towards this conjecture. Finally, we give an overview of the proof of a joint work by Ziyang Gao, Vesselin Dimitrov, and the author towards a question of Mazur regarding upper bounds for the cardinality #C

(Q)

. The upper bound will depend on the genus

g

and the Mordell-Weil rank of the Jacobian of

C

. For a special case of this result that does not make reference to Jacobians, we refer to Section 6.

1.1. The Mordell Conjecture

We begin by recalling Faltings's Theorem [25], a finiteness statement originally conjectured by Mordell [48]. By a curve we mean a geometrically irreducible projective variety of dimension 1. Throughout, we let

F

denote a number field and

\bar{F}

a fixed algebraic closure of

F

Theorem 1.1 (Faltings [25]). Let

C

be a smooth curve of genus at least 2 defined over a number field

F

. Then

C (F)

is finite.

If the genus of

C

is small, then one cannot expect finiteness. Indeed, the set

C (F)

is nonempty after replacing

F

by a suitable finite extension. If

C

has genus 0 , then

C

is isomorphic to the projective line and thus

C (F)

is infinite. If

C

has genus 1, then

C

together with a point in

C (F)

is an elliptic curve. In particular, we obtain an algebraic group. After possibly extending

F

again, we may assume that

C (F)

contains a point of infinite order. So

C (F)

is infinite.

To prove the Mordell Conjecture, Faltings first proved the Shafarevich Conjecture for abelian varieties. At the time, the latter was known to imply the Mordell Conjecture thanks to a construction of Kodaira-Parshin.

Later, Vojta [62] gave a different proof of the Mordell Conjecture that is rooted in diophantine approximation. Bombieri [8] then simplified Vojta's proof. We will recall Vojta's
approach for curves in Section 3. The technical heart is the Vojta inequality which we formulate below as Theorem 3.1.

Faltings generalized Vojta's proof of the Mordell Conjecture to cover subvarieties of any dimension of an abelian variety. Indeed, Faltings [26, 27] and Hindry [36] proved the Mordell-Lang Conjecture for subvarieties of abelian varieties. Let

A

be an abelian variety defined over

F

and suppose

Γ

is a subgroup of

A (\bar{F})

. The division closure of

Γ

is the subgroup

{P \in A (\bar{F}) : there exists an integer n \geq 1 with n P \in Γ}

A (\bar{F})

. For example, the division closure of the trivial subgroup

Γ = {0}

is the subgroup

A_{tors}

of all points of finite order of

A (\bar{F})

. The following theorem holds for all base fields of characteristic 0 .

Theorem 1.2 (Mordell-Lang conjecture, Faltings, Hindry). Let

A

be an abelian variety defined over

F

and let

Γ

be the division closure of a finitely generated subgroup of

A (\bar{F})

. If

V

is an irreducible closed subvariety of

A

, then the Zariski closure of

V (\bar{F}) \cap Γ

V

is a finite union of translates of algebraic subgroups of

A

The special case when

Γ = A_{tors}

is called the Manin-Mumford Conjecture and was proved by Raynaud [53].

More recently, Lawrence and Venkatesh [41] gave yet another proof of the Mordell Conjecture. It was inspired by Faltings's original approach and the method of ChabautyKim. We refer to the survey [6] on these developments.

In this survey we concentrate mainly on the case of curves and comment on possible extensions to the higher dimensional case.

1.2. Some remarks on effectivity

Despite the variety of approaches to the Mordell Conjecture, no effective proof is known. For example, if the curve

C

is presented explicitly as the vanishing locus of homogeneous polynomial equations with rational coefficients, say, then in full generality we know no algorithm that produces the finite list of rational points of

C

. The question of effectivity is already open in genus 2 , for example, for the family

Y^{2} = X^{5} + t

parametrized by

t

. Proving an effective version of the Mordell Conjecture is among the most important outstanding problems in diophantine geometry.

Although no general algorithm that determines the set of rational points is currently known, it is sometimes possible to determine the set of rational points. For example, we refer to the Chabauty-Coleman method

[13, 15]

which provides a clean upper bound for the number of rational points subject to a hypothesis on the Mordell-Weil rank of the Jacobian of

C

. In several applications, this bound equals a lower bound for the number of rational points coming from a list of known rational points. Moreover, aspects of Kim's generalization of the Chabauty method were used by Balakrishnan, Dogra, MÃ¼ller, Tuitman, and Vonk [5] to compute all rational points of the split Cartan modular curve of level 13 which appears in relation to Serre's uniformity question. A different approach motivated by work of Dem-
janenko and the theory of unlikely intersections was developed in a program by Checcoli, Veneziano, Viada [14]. Here too a condition on the rank of the curve's Jacobian is required for the method to apply. An remarkable aspect to this approach is that the authors obtain an explicit upper bound for the height of a rational point.

1.3. The number of rational points: conjectures and results

Given

C

and

F

as in Theorem 1.1, which invariants of

C

need to appear in an upper bound for #C(F)?

Example 1.3. (i) Consider the hyperelliptic curve

C

presented by

y^{2} = (x - 1) \dots (x - 2022)

Its genus equals

(2022 - 2) / 2 = 1010

. Then

C

contains the rational points

(1, 0), \dots, (2022, 0)

. Together with the two points at infinity, we obtain at least 2024 rational points. This example easily generalizes to higher genus. For any

g \geq 2

and square-free

f \in Q [X]

of degree

2 g + 2

, the equation

y^{2} = f (x)

determines a hyperelliptic curve

C

of genus

g

. If

f

splits into (pairwise distinct) linear factors over

Q

, then

# C (Q) \geq 2 g + 4

. So any upper bound for

# C (Q)

must depend on the genus.

This lower bound is far from the truth. Stoll discovered a genus 2 curve defined over

Q

with 642 rational points in a family of such curves constructed by Elkies. Mestre showed that for all

g \geq 2

there is a smooth curve of genus

g

defined over

Q

with at least

8 g + 16

rational points.

(ii) Let us now fix the curve

C

and let the number field

F

vary. We take

C

as the genus 2 hyperelliptic curve presented by

y^{2} = x^{5} + 1

. Consider an integer

n \geq 0

and the points

{(m, \pm {(m^{5} + 1)}^{1 / 2}) : m \in {0, \dots, n}}

. So

C (F)

has at least

2 n + 2 + 1

elements where

F = Q {({(m^{5} + 1)}^{1 / 2})}_{m \in {1, \dots, n}}

. Any upper bound

C (F)

, even for

C

fixed, must depend on

F

Gabriel Dill pointed out that the number of

F

-points grows at least logarithmically in the degree

[F : Q]

in this case. Indeed,

[F : Q] \leq 2^{n}

, so #C

F (F) \geq 2 n \geq

2 (\log [F : Q]) / \log 2

Let us consider the modular curve

X_{0} (37)

which has genus 2 and is defined over

Q

. Let

p

be one of the infinitely many prime numbers for which the Legendre symbol satisfies

(- p / 37) = 1

; so 37 splits in the quadratic field

K = Q (\sqrt{- p})

. Let

F

denote the Hilbert Class Field of

K

. There is an elliptic curve

E

defined over

F

with complex multiplication by the ring of integers of

K

. Moreover,

E

admits an isogeny of degree 37 to an elliptic curve defined over

F

. Thus

X_{0}

(37) has an

F

-rational point. The Galois group

Gal (F / K)

acts on the

F

-rational points of

X_{0} (37)

. It is also known to act transitively on the moduli of elliptic curves with the same endomorphism ring as

E

. Thus

# X_{0} (37) (F)

is no less than

[F : K]

which equals the class number of

K

by Class Field Theory. So

# X_{0} (37) (F) \geq [F : K] = [F : Q] / 2

. By the Landau-

Siegel Theorem,

[F : Q] \to \infty

p \to \infty

. In particular, any upper bound for

# X_{0} (37) (F)

must grow at least linearly in

[F : Q]

The Uniformity Conjecture by Caporaso-Harris-Mazur [11] predicts that the genus and base field of a curve are the only invariants required for a general upper bound.

Conjecture 1.4 (Caporaso-Harris-Mazur). Let

g \geq 2

be an integer and

F

a number field. There exists

c (g, F) \geq 1

such that if

C

is a smooth curve of genus

g

defined over

F

, then

# C (F) \leq c (g, F)

Caporaso, Harris, and Mazur showed that the Uniformity Conjecture follows from the Weak Lang Conjecture which is an extension of the Mordell Conjecture to higher dimension. It states that if

V

is a smooth projective variety defined over

F

of positive dimension and general type, then

V (F)

is not Zariski dense in

V

. Pacelli [50] showed that #C(F) is bounded from above in function of

g

and

[F : Q]

under the Weak Lang Conjecture after Abramovich [1] treated the case of quadratic and cubic extensions earlier. A refined version of the Weak Lang Conjecture implies, again by work of Caporaso-Harris-Mazur, that #C(F) can be bounded from above in function of the genus, if we omit finitely many

F

-isomorphism classes of

C

, see also [12] for a correction. RÃ©mond [56] has evidence towards this stronger version of the Uniformity Conjecture. Alpoge [2] showed that, on average, a smooth curve of genus 2 defined over

Q

has a uniformly bounded number of rational points.

Mazur [46] posed the following question, which is a weaker version of the Uniformity Conjecture. We let

Jac (C)

denote the Jacobian of a smooth curve

C

defined over a field. Then

Jac (C)

is a principally polarized abelian variety whose dimension equals the genus of

C

. If the base field is a number field

F

, then

Jac (C) (F)

is a finitely generated abelian group by the Mordell-Weil Theorem.

Question 1.5 (Mazur [46, P. 223]). Let

g \geq 2

and

r

be integers and let

F

be a number field. There exists

c (g, r, F) \geq 1

such that if

C

is a smooth curve of genus

g

defined over

F

such that the rank of the Mordell-Weil group satisfies rk

Jac (C) (F) \leq r

, then

# C (F) \leq c (g, r, F)

Let us review some work on upper bounds for

# C (F)

. Parshin [59] showed how to extract an upper bound for the number of rational points from Faltings's theorem. In his original paper, Vojta [62] gave a blueprint on how to bound from above the number of rational points for a general

C

. This bound was refined by Bombieri [8] and de Diego [19]. However, these works did not provide an answer to Mazur's question.

To formulate de Diego's results we need some additional notation. We also require the Weil height on projective space and the NÃ©ron-Tate (or canonical) height, they are both defined in Section 2. Let

S

be an irreducible, smooth, quasiprojective variety defined over a number field

F

and presented with an immersion

S \subseteq P^{n}

defined over

F

. De Diego's Theorem holds for a smooth family of curves parametrized by the base

S

. Indeed, let

C \to S

be a smooth morphism such that each fiber is a smooth curve of genus

g \geq 2

. We write

C_{s}

for the fiber of

C \to S

above

s \in S (\bar{F})

. This is a smooth curve defined over

\bar{F}

. Let

K_{s}

denote the canonical class on

C_{S}

, we identify it with a divisor class modulo linear equivalence of
degree

2 g - 2

. If

P \in C_{s} (\bar{F})

, then

(2 g - 2) [P] - K_{s}

is well defined as a divisor class of degree 0 . So it represents a point in

Jac (C_{s})

. In this way we obtain a morphism

j_{s} : C_{s} \to Jac (C_{s}) given by P \mapsto ((2 g - 2) [P] - K_{s}) / \sim

Let

θ_{s}

denote the theta divisor on

Jac (C_{s})

and

{\hat{h}}_{s} = {\hat{h}}_{C_{s}, θ_{s}}

the canonical height on

Jac (C_{s})

attached to this divisor.

Theorem 1.6 (de Diego [19]). There exists

c (C) > 1

such that if

F^{'} / F

is a finite extension and

s \in S (F^{'})

, then

# {P \in C_{s} (F^{'}) : {\hat{h}}_{s} (j_{s} (P)) \geq c (C) (1 + h (s))}} \leq \frac{55}{2} \cdot 7^{r k J a c (C) (F^{'})}

Roughly speaking, this theorem tells us that the number of points of

C_{s}

of sufficiently large canonical height is bounded as in Mazur's question. We will often call these points large points. It is striking that the constant 7 is admissible for all genera; a fact that already appeared in Bombieri's work [8]. For smooth curves of genus 2 defined over

Q

with a marked Weierstrass point, Alpoge [2] improved 7 to 1.872 .

Observe that

\begin{matrix} (1.1) & {P \in C_{s} (F^{'}) : {\hat{h}}_{s} (j_{s} (P)) < c (C) (1 + h (s))} \end{matrix}

is finite by the Northcott property for height functions which we will review in Section 2. To obtain a positive answer to Mazur's question we need, roughly speaking, to get a similar bound as in Theorem 1.6 for the cardinality of (1.1). There are quantitative versions of Northcott's Theorem. Estimating the cardinality (1.1) with these does, however, introduce a dependence on

h (s)

c (C)

. Indeed, David and Philippon proved a lower bound for the canonical height that, when combined with RÃ©mond's explicit version of the Vojta inequality, yields the next theorem. To state their result, we momentarily shift our focus from families of smooth curves and their Jacobians to a curve immersed in an abelian variety.

g

-dimensional principally polarized abelian variety defined over

F

. Let

Γ

be the division closure of a finitely generated subgroup of

A (\bar{F})

of rank

r

and let

C \subseteq A

be a curve that is not smooth of genus 1 . Then

C (\bar{F}) \cap Γ

is finite of cardinality at most

{(2^{34} h_{0} (A) \deg C)}^{g^{20} (r + 1)}

Here

\deg C

is the degree of

C

with respect to the principal polarization. Moreover,

h_{0} (A)

is a height of the abelian variety

A

whose definition involves classical theta functions and the degree

[F : Q]

. Roughly speaking,

h_{0} (A)

encodes a bound for the coefficients needed to reconstruct the abelian variety

A

. Mazur's question does not allow for a dependence on

h_{0} (A)

. The hypothesis that

C

is not smooth of genus 1 is natural and cannot be dropped in general. It is equivalent to stating that

C

is not a translate of an algebraic subgroup of

A

David and Philippon's contribution to Theorem 1.7 was their lower bound for the canonical height, see [17, THÃ‰ORÃˆME 1.4]. RÃ©mond [54] made Vojta's approach (and an inequality of Mumford) completely explicit. David-Philippon and RÃ©mond have a result for subvarieties of

A

of any dimension. In other words, they provide an explicit version of the Mordell-Lang Conjecture.

David and Philippon's approach to Mazur's question and its higher-dimensional counterparts is via a strong quantitative version of the Bogomolov Conjecture on points of small height. A suitable version is Conjecture 1.5 [18] where the lower bound for the canonical height grows linearly in the Faltings height. We refer to [18, THÃ‰ORÄ–ME 1.11] regarding the connection to rational points and more generally the Mordell-Lang Conjecture.

David and Philippon were able to strengthen their height lower bound when

A

is a power of an elliptic curve. This provided more evidence towards a positive answer for Mazur's question. Here is a version of their result for curves; their general result holds for subvarieties of a power of an elliptic curve.

Theorem 1.8 (David and Philippon [18, THÃ‰oRÃˆme 1.13]). Let E be an elliptic curve defined over

F

and let

g \geq 2

be an integer. Suppose

Γ

is the division closure of a finitely generated subgroup of

E^{g} (\bar{F})

of rank

r

. If

C \subseteq E^{g}

is a curve that is not smooth of genus 1 , then

# C (\bar{F}) \cap Γ \leq \deg (C)^{7 g^{18} (1 + r)}

Thanks to a specialization argument, David and Philippon extended the above result to include the case where

F

is an arbitrary field of characteristic 0 . David, Nakamaye, and Philippon [16] then proved the existence of a

(g - 2)

-dimensional family of curves of genus

g

for which Mazur's question has a positive answer.

We now very briefly turn to some cardinality estimates using the Chabauty-Coleman method, which is based on

p

-adic analysis. It can produce finiteness of

C (F)

with a clean cardinality estimate subject to a restriction on the rank of the Mordell-Weil group.

Theorem 1.9 (Coleman [15]). Suppose

C

is a smooth curve of genus

g \geq 2

defined over

Q

with

rkac (C) (Q) \leq g - 1

. If

p > 2 g

is a prime number where

C

has good reduction

\tilde{C}

, then

# C (Q) \leq 2 g - 2 + # \tilde{C} (F_{p})

In combination with the Hasse-Weil bound

# \tilde{C} (F_{p}) \leq p + 1 + 2 g \sqrt{p}

, the estimate above yields a bound for

# C (Q)

in terms of

g

and

p

alone. Observe that a dependence in the arithmetic of

C

appears through the prime

p

. Stoll was able to remove this dependence for hyperelliptic curves at the cost of a stronger restriction on the rank of the Mordell-Weil group.

Theorem 1.10 (Stoll [58]). Let

g \geq 2

and

d \geq 1

be integers. There exists

c (g, d) > 0

with the following property. Suppose

C

is a smooth hyperelliptic curve of genus

g

defined over

F

with

[F : Q] \leq d

. If

rk Jac (C) (Q) \leq g - 3

, then

# C (F) \leq c (g, d)

Later, Katz, Rabinoff, and Zureick-Brown dropped the hyperellipticity condition.

Theorem 1.11 (Katz, Rabinoff, and Zureick-Brown [38]). Let

g \geq 2

and

d \geq 1

be integers. There exists

c (g, d) > 0

with the following property. Suppose

C

is a smooth curve of genus

g

defined over

F

with

[F : Q] \leq d

. If

rk Jac (C) (Q) \leq g - 3

, then

# C (F) \leq c (g, d)

After this detour to the Chabauty-Coleman method, we return to Vojta's method. Vesselin Dimitrov, Ziyang Gao, and the author have recently proved a lower bound for the canonical height that can be used as a replacement for the lower bounds by David and Philippon

[17, 18]

in the context of Mordell's Conjecture. We recall this height inequality in Section 4.2 below. Indeed, it led to a positive answer to a strengthening of Mazur's question. The following result is new already in genus 2 .

Theorem 1.12 (Dimitrov, Gao, and Habegger [24, theorem 1.1]). Let

g \geq 2

and

d \geq 1

be integers, there exist

c^{'} (g, d) > 1

and

c (g, d) > 1

with the following property. Suppose

C

is a smooth curve of genus

g

defined over a number field

F

such that

[F : Q] \leq d

. Then

# C (F) \leq c^{'} (g, d) \cdot c (g, d)^{r k Jac (C) (F)}

Regarding the Caporaso-Harris-Mazur Uniformity Conjecture, we ask

Question 1.13. Can the cardinality

# C (F)

be bounded from above by a function that is polynomial in

[F : Q]

and

g

No one currently knows an algorithm that computes the rank of the Mordell-Weil group

Jac (C) (F)

. However, upper bounds for this rank follow, for example, from the OoeTop Theorem [49]. We discuss this in more depth in Section 6.

Our results also cover points on

C

that lie in the division closure of a finitely generated subgroup. Let

\bar{Q}

denote the algebraic closure of

Q

C

. The Jacobian Jac

(C)

of a smooth curve

C

of genus

g

defined over

\bar{Q}

corresponds to a

\bar{Q}

-point of the coarse moduli space

A_{g}

g

-dimensional principally polarized abelian varieties. We let

[Jac (C)]

denote the point of

A_{g} (\bar{Q})

corresponding to

Jac (C)

with its canonical principal polarization.

For example, if

g = 1

then

A_{g} = A^{1}

is the affine line. If

E

is an elliptic curve defined over

\bar{Q}

, then

[E]

is the

j

-invariant of

E

In general,

A_{g}

is a quasiprojective variety of dimension

g (g + 1) / 2

defined over

Q

. We may fix an immersion

ι : A_{g} ↪ P^{n}

into projective space. Then the absolute logarithmic Weil height

h

, see Section 2 for a definition, pulls back to a function

h \circ ι : A_{g} (\bar{Q}) \to [0, \infty)

C

is a smooth curve of genus

g \geq 1

defined over

\bar{Q}

and if

P_{0} \in C (\bar{Q})

, then a point

P \in C (\bar{Q})

defines a divisor

[P] - [P_{0}]

of degree 0 . One obtains a closed immersion

C ↪ Jac (C) from P \mapsto ([P] - [P_{0}]) / \sim

where

\sim

again denotes linear equivalence, induces a closed immersion. We will write

C - P_{0}

for the image of

C

Jac (C)

Theorem 1.14 (Dimitrov, Gao, and Habegger [24, THEOREM 1.2]). Let

g \geq 2

be an integer. There exist

c (g, ι) > 1, c^{'} (g, ι) > 0

, and

c^{''} (g, ι) > 0

that depend on

g

and the immersion

ι

with the following property. Suppose

C

is a smooth curve of genus

g

defined over

\bar{Q}

and let

P_{0} \in C (\bar{Q})

. Let

Γ

be the division closure of a finitely generated subgroup of

Jac (C) (\bar{Q})

of rank

r

. If

h (ι ([Jac (C)])) \geq c^{''} (g, ι) then # (C - P_{0}) (\bar{Q}) \cap Γ \leq c^{'} (g, ι) c (g, ι)^{r}

In particular, we may take

Γ = Jac (C)_{tors}

and

r = 0

. Thus the theorem yields a uniform bound for the number of torsion points that lie on

C - P_{0}

if the height of

ι ([Jac (C)])

is sufficiently large.

Suppose that

C

is defined over a number field

F

. Then

[Jac (C)]

is an

F

-rational point of the moduli space

A_{g}

. If we impose also

h (ι ([Jac (C)])) < c^{''} (g, ι)

, then

[Jac (C)]

lies in a finite set by the Northcott property. Thus

Jac (C)

is in one of at most finitely many

\bar{Q}

-isomorphism classes and so is

C

by the Torelli Theorem.

Raynaud proved the following result which is the Manin-Mumford Conjecture for curves.

Theorem 1.15 (Raynaud [52]). Let

C

be smooth curve defined over

C

of genus at least 2 . Then

(C - P_{0}) \cap Jac (C)_{tors}

is finite.

Theorem 1.14 gives evidence towards the Uniform Manin-Mumford Conjecture which states that

(C - P_{0}) \cap Jac (C)_{tors}

is bounded from above in terms of the genus

g

only for any smooth curve

C

of genus

g \geq 2

defined over any field in characteristic 0 .

Using a different approach involving equidistribution and motivated by dynamical systems, DeMarco, Krieger, and Ye [20] had made substantial progress towards the Uniform Manin-Mumford Conjecture. They proved it for smooth curves of genus 2 defined over

C

that are double covers of an elliptic curve when the base point

P_{0}

is a Weierstrass point.

In a preprint, KÃ¼hne [39] complemented the method in [24] using ideas from equidistribution to prove the Uniform Manin-Mumford Conjecture.

Theorem 1.16 (KÃ¼hne [39]). Let

g \geq 2

be an integer. There exist

c (g) > 1

and

c^{'} (g) > 1

that depend on

g

with the following property. Suppose

C

is a smooth curve of genus

g

defined over

C

and let

P_{0} \in C (C)

. Let

Γ

be the division closure of a finitely generated subgroup of

Jac (C) (C)

of rank

r

. Then

# (C - P_{0}) (C) \cap Γ \leq c^{'} (g) c (g)^{r}

In contrast to Theorem 1.14, KÃ¼hne is able to handle curves

C

defined over

\bar{Q}

for which

[Jac (C)]

has height less than

c^{''} (g, ι)

. Once uniformity is established for all curves over

\bar{Q}

, KÃ¼hne is able to pass to the base field

C

using a specialization argument laid out by Dimitrov, Gao, and the author [22] which relies on a result of Masser [43]. KÃ¼hne thus answers an older question of Mazur, see the top of page 234 [45], and obtains the full Mordell-Lang variant for curves.

DeMarco and Mavraki's [21] work on a relative version of the Bogomolov conjecture, see [72] and [22], motivated KÃ¼hne [39,40] to extend the reach of Arakelovian equidistribution methods of Szpiro-Ullmo-Zhang [60] and Yuan [65] to families of abelian varieties over a quasiprojective base. For algebraic curves, this settles the uniform Bogomolov and the uniform Manin-Mumford conjectures.

Yuan [66] recently gave another proof of Theorem 1.16. His method also runs via a uniform Bogomolov theorem and thus contains aspects related to height lower bounds. However, Yuan's approach relies on arithmetic bigness, rather than on equidistribution. It is independent of the approaches mentioned above and uses a new theory of adelic line bundles over quasiprojective varieties developed by Yuan and Zhang [67] which generalizes Zhang's theory [70] in the projective case. They derive a height inequality for a polarized dynamical system, see Theorem 1.3.2 and Section 6 [67], that extends our own bound. One aspect of Yuan's method is that it works for global fields in any characteristic.

We come to some questions regarding the base constant

c (g)

in the estimates above. In the context of Mordell's Conjecture, Bombieri observed that the number of large points is bounded by a multiple of

7^{r k J a c} (C) (F)

Question 1.17. Can the base 7 in the estimate for the number of large points as in Theorem 1.6 be replaced by a function in

g

that tends to 1 for

g \to \infty

Alpoge [2] used the Kabatiansky-Levenshtein estimates on spherical codes to improve on the constant 7 in genus 2. It is quite possible that Alpoge's approach will shed light on this last question.

Concerning the constant

c (g)

in Theorem 1.16, we pose the following two questions which also cover the moderate, i.e., nonlarge, points. They were inspired by questions of Helfgott.

Question 1.18. Can we choose the

c (g)

in Theorem 1.16 such that there exists

B \geq 1

with

c (g) \leq B

for all integers

g \geq 2

Question 1.19. Can we choose the

c (g)

in Theorem 1.16 with

lim_{g \to \infty} c (g) = 1

Recently, Gao, Ge, and KÃ¼hne [32] completed the proof of the Uniform MordellLang Conjecture for a subvariety

V

of a polarized abelian variety

A

of any dimension. Uniformity here amounts to bounding the number of irreducible components of the Zariski closure in Theorem 1.2 from above by

c^{'} (\dim A, \deg V) c (\dim A, \deg V)^{r}

. Their result holds over all base fields in characteristic 0 .

We refer to the comprehensive survey by Gao [31] that gives an overview of these recent developments and how they are interlinked.

Here is a brief overview of this survey. In Section 2 we recall some fundamental properties of two height functions: the Weil and NÃ©ron-Tate heights. They play a central role in the proof of Theorem 1.12. Then in Section 3 we describe Vojta's approach to the Mordell Conjecture. Later we return to the two height functions and describe their interactions on a family of abelian varieties. This is done in Section 4. Here we also describe the Betti map, an important analytic tool. In Section 5 we sketch how all this fits together in the proof of Theorem 1.12. In the final section we discuss an estimate for the number of rational points on a hyperelliptic curve that does not make reference to Jacobians.

2. HEIGHTS

Height functions are at the heart of Vojta's proof of the Mordell Conjecture and subsequent results such as Theorem 1.12. We will review two flavors of heights. The first one is the absolute logarithmic Weil height which is defined on algebraic points of the projective space. One can also use it to define a class of height functions on a projective variety equipped with an invertible sheaf. The second height function is the canonical or NÃ©ron-Tate height on an abelian variety, also equipped with an invertible sheaf. The latter is compatible with the group structure on the abelian variety.

2.1. The absolute logarithmic Weil height

We review here briefly the main properties of the Weil height. For a thorough treatment, we refer to [9, CHAPTERS 1 AND 2] or [37, PART B].

We begin by defining the height of a rational point on projective space

P^{n}

Definition 2.1. Let

P \in P^{n} (Q)

. There exist projective coordinates

(x_{0}, \dots, x_{n}) \in

Z^{n + 1} ∖ {0}

P = [x_{0} : \dots : x_{n}]

with

\gcd (x_{0}, \dots, x_{n}) = 1

. Then we set

h (P) = \log max {| x_{0} |, \dots, | x_{n} |}

The vector

(x_{0}, \dots, x_{n})

is uniquely determined up to a sign, and so

h (P)

is well defined. For example,

h ([2 : 4 : 6]) = h ([1 : 2 : 3]) = h ([1 / 3 : 2 / 3 : 1]) = \log 3

The following theorem is a straightforward consequence of the definition of the Weil height.

Theorem 2.2 (Northcott property). The set

{P \in P^{n} (Q) : h (P) \leq B}

is finite for all

B

Defining the height of an algebraic point in

P^{n} (\bar{Q})

requires some basic algebraic number theory. Indeed, let

K

be a number field. We let

M_{K}

denote the set of absolute values

| \cdot | : K \to [0, \infty)

that extend either the standard absolute value on

Q

or a

p

-adic absolute value for some prime

p

. Then

M_{K}

is called the set of places of

K

. For each

v \in M_{K}

, one sets

d_{v} = [K_{v} : Q_{w}]

where

K_{v}

is a completion of

K

with respect to

v

and

Q_{w}

is the completion of

Q

K_{v}

with respect to

w = {v |}_{Q}

Definition 2.3. Let

P \in P^{n} (\bar{Q})

and let

K

be a number field such that

P = [x_{0} : \dots : x_{n}]

where

(x_{0}, \dots, x_{n}) \in K^{n + 1} ∖ {0}

. The absolute logarithmic Weil height, or just Weil height, is

\begin{matrix} (2.1) & h (P) = \frac{1}{[K : Q]} \sum_{v \in M_{K}} d_{v} \log max {{| x_{0} |}_{v}, \dots, {| x_{n} |}_{v}} \end{matrix}

The normalization constants

d_{v}

are chosen such that the product formula

\prod_{v \in M_{K}} | x |_{v}^{d_{v}} = 1

holds for all

x \in K ∖ {0}

. This guarantees that the right-hand side of (2.1) is independent of the choice of projective coordinates of

P

. In particular, we may assume that some projective coordinate of

P

equals 1. Thus

h (P) \geq 0

for all

P \in P^{n} (\bar{Q})

. Moreover,

h (P)

is independent of the field

K

containing the projective coordinates.

For applications to diophantine geometry, it is useful to have a height function defined on algebraic points of an irreducible projective variety

V

defined over

\bar{Q}

. But without additional data there is no reasonable way to define a height on

V (\bar{Q})

However, if

V

is a subvariety of the projective space

P^{n}

, then we may restrict the Weil height

h : P^{n} (\bar{Q}) \to R

to a function

V (\bar{Q}) \to R

. Slightly more generally, if

V \to P^{n}

is an immersion, then we may pull back the Weil height to

V (\bar{Q})

Recall that an immersion

V \to P^{n}

is induced by a tuple of

(n + 1)

global sections of a very ample invertible sheaf on

V

. Conversely, given a very ample invertible sheaf

L

V

, we can fix a basis of the vector space of global sections of

L

and obtain an immersion

L : V \to P^{n}

. So we obtain a function

h \circ L : V (\bar{Q}) \to [0, \infty)

. There is a wrinkle here, this function depends not only on

(V, L)

but also on the basis of the vector space of global sections. A different basis will lead to a function

V (\bar{Q}) \to [0, \infty)

that differs from

h \circ L

by a bounded function on

V (\bar{Q})

. We define

h_{V, L}

to be the equivalence class of functions

V (\bar{Q}) \to R

modulo bounded functions that contains

h \circ ι L

L

is an ample invertible sheaf on

V

, then there exists an integer

n \geq 1

such that

L^{\otimes n}

is very ample. We then define

h_{V, L} = \frac{1}{n} h_{V, L \otimes n}

; this is again only defined up to a bounded function on

V (\bar{Q})

. The equivalence class does not depend on the choice of

n

Finally, an arbitrary invertible sheaf

L

in the Picard group

Pic (V)

V

is of the form

F \otimes M^{\otimes (- 1)}

with

F

and

M

ample on

V

. The difference

h_{V, F} - h_{V, M}

is well defined up to a bounded function on

V (\bar{Q})

. It does not depend on the pair

F, M

with difference

L

, and we denote it by

h_{V, L}

. It is called the Weil height attached to

(V, L)

Theorem 2.4. Let us keep the notation above. In particular,

V

is an irreducible projective variety defined over

\bar{Q}

(i) The association

L \mapsto h_{V, L}

is a group homomorphism with target the group of real-valued maps

V (\bar{Q}) \to R

modulo bounded functions.

(ii) For

V

equal to projective space and

L

the hyperplane bundle

O (1)

, the Weil height from Definition 2.3 represents

h_{P^{n}, O (1)}

(iii) Suppose

W

is a further irreducible projective variety defined over

\bar{Q}

and

f

W \to V

is a morphism. For all

L \in Pic (V)

we have

h_{V, L} \circ f = h_{W, f} * L

. As usual, this equality is understood as an equality of equivalence classes of functions.

(iv) Suppose

L \in Pic (V)

admits a nonzero global section

s

. Then

h_{V, L}

is bounded from below on the complement of the vanishing locus of

s

. In particular,

h_{V, L}

is bounded from below on a Zariski open and dense subset of

V

Suppose that

V

is defined over a number field

F \subseteq \bar{Q}

and

L \in Pic (V)

is ample. Then the Northcott property holds for points of bounded degree, i.e.,

{P \in V (\bar{F}) : h_{V, L}^{'} (P) \leq B and [F (P) : F] \leq D}

is finite where

h_{V, L}^{'}

denotes any representative of

h_{V, L}

Let

V

be an irreducible projective variety defined over

\bar{Q}

. We conclude this section by discussing a powerful tool to translate geometric information, here on intersection numbers, into an inequality of heights. The basic question is the following. Given invertible sheaves

F

and

M

V

, under what conditions can one bound

h_{V, M}

from above in terms of

h_{V, F}

(i) We first consider the special case

V = P^{n}

. As

Pic (P^{n})

is isomorphic to

Z

, any Weil height is some integral multiple of

h_{P^{n}, O (1)}

. So

h_{V, F}

and

h_{V, M}

are

Z

-linearly dependent.

(ii) Let us again suppose that

V

is general and that

F

is ample. Then there exists an integer

k \geq 1

such that

F \otimes k \otimes M^{\otimes (- 1)}

is ample. So for some positive integer

l \geq 1

the power

F^{\otimes k l} \otimes M^{\otimes (- l)}

is very ample. In particular, it admits a global section that does not vanish at a prescribed point of

V (\bar{Q})

. Theorem 2.4, parts (i) and (iv), imply

k l h_{V, F} - l h_{V, M} = h_{V, F \otimes k l \otimes M \otimes (- l)} \geq 0

this must be parsed as an inequality between functions on

V (\bar{Q})

defined up to addition of a bounded function. We conclude

\begin{matrix} (2.2) & h_{V, M} \leq k h_{V, F} \end{matrix}

(iii) For some applications, such as Theorem 1.12, the ampleness hypothesis on

F

in (i) is not flexible enough. Moreover, we would like some way to estimate the factor

k

in (2.2) from above. We now describe a criterion of Siu that provides a solution to these two issues.

An invertible sheaf

L \in Pic (V)

is called big if

\underset{k \to \infty}{lim inf} \frac{\dim H^{0} (V, L^{\otimes k})}{k^{\dim V}} > 0

here

H^{0} (V, L)

denotes the vector space of global sections of

L

L

is a big invertible sheaf, then

L^{\otimes k}

has a nonzero global section for some

k \geq 1

. Then using (i) and (iv) of Theorem 2.4 we see that

h_{V, L} = \frac{1}{k} h_{V, L \otimes k}

is bounded from below on a Zariski open and dense subset of

V

For example, if

L = F \otimes M^{\otimes (- 1)}

is big, then, again by Theorem 2.4(i), we find

h_{V, F} \geq h_{V, M}

on a Zariski open and dense subset of

V

We now come Siu's Criterion; it ensures that

F \otimes M^{\otimes (- 1)}

is big. An invertible sheaf

L \in Pic (V)

is called nef, or numerically effective, if

(L \cdot [C]) \geq 0

for all irreducible curves

C \subseteq V

Siu's Criterion requires that

F

and

M

are both nef and that the intersection numbers on

V

satisfy

(F \cdot \dim V) > (\dim V) (F \cdot (\dim V - 1) \cdot M)

. With these hypotheses

F \otimes M^{\otimes (- 1)}

is big; see [42, THEOREM 2.2.15].

Say

F

and

M

are nef and

(F \cdot \dim V) > 0

. Let

k

and

l

be positive integers with

(\dim V) \frac{(F \cdot (\dim V - 1) \cdot M)}{(F \cdot \dim V)} < \frac{k}{l}

then

F^{\otimes k} \otimes M^{\otimes (- l)}

is big. So

{h_{V, M} |}_{U} \leq {\frac{k}{l} h_{V, F} |}_{U}

holds on some Zariski open and dense

U \subseteq V

This allows us to compare the heights

h_{V, M}

and

h_{V, F}

if we have information on the intersection numbers, at least on a rather large subset of

V (\bar{Q})

Yuan [65] proved an arithmetic version of this criterion in his work on equidistribution. The author [34] used Siu's Criterion to study unlikely intersections in abelian varieties.

2.2. The canonical height on an abelian variety

Let

F \subseteq \bar{Q}

be a number field and

A

an abelian variety defined over

F

. If

L

6.4. Eisenstein series to cusp forms

6.5. Cusp forms to Galois representations

6.6. Galois representations to Galois cohomology classes

6.7. Galois cohomology classes to class groups

7. EXPLICIT FORMULA FOR BRUMER-STARK UNITS

7.1. Shintani's method

Theorem 7.2. The formal linear combination

7.2. The formula

7.3. Horizontal Iwasawa theory

7.4. The Greenberg-Stevens L L L\mathscr{L}L-invariant

7.5. The method of Darmon-Pozzi-Vonk

ACKNOWLEDGMENTS

FUNDING

REFERENCES

SAMIT DASGUPTA

MAHESH KAKDE

ARITHMETIC AND DYNAMICS ON VARIETIES OF MARKOFF TYPE

ALEXANDER GAMBURD

ABSTRACT

MATHEMATICS SUBJECT CLASSIFICATION 2020

KEYWORDS

1. INTRODUCTION

FIGURE 1

FIGURE 2

2. THE UNREASONABLE(?) UBIQUITY OF MARKOFF EQUATION

3. STRONG APPROXIMATION

4. REAL DYNAMICS ON SURFACES OF MARKOFF TYPE

FIGURE 5

FIGURE 7

FIGURE 8

FIGURE 9

FIGURE 12

FIGURE 13

5. AN ASYMPTOTIC FORMULA FOR INTEGER POINTS ON MARKOFF-HURWITZ VARIETIES

6. HASSE PRINCIPLE ON SURFACES OF MARKOFF TYPE

FIGURE 17

FIGURE 19

ACKNOWLEDGMENTS

FUNDING

REFERENCES

ALEXANDER GAMBURD

THE NUMBER OF RATIONAL POINTS ON A CURVE OF GENUS AT LEAST TWO

PHILIPP HABEGGER

ABSTRACT

MATHEMATICS SUBJECT CLASSIFICATION 2020

KEYWORDS

1. INTRODUCTION

1.1. The Mordell Conjecture

1.2. Some remarks on effectivity

1.3. The number of rational points: conjectures and results

2. HEIGHTS

2.1. The absolute logarithmic Weil height

2.2. The canonical height on an abelian variety

7.4. The Greenberg-Stevens $L$ -invariant